Mathematics links epistemology to ontology, but it’s not that simple

A recurring theme on this blog is the relationship between mathematics and reality. It started with the Pythagoreans (in Western philosophy) and was famously elaborated upon by Plato. I also think it’s the key element of Kant’s a priori category in his marriage of analytical philosophy and empiricism, though it’s rarely articulated that way.
 
I not-so-recently wrote a post about the tendency to reify mathematical objects into physical objects, and some may validly claim that I am guilty of that. In particular, I found a passage by Freeman Dyson who warns specifically about doing that with Schrodinger’s wave function (Ψ, the Greek letter, psi, pronounced sy). The point is that psi is one of the most fundamental concepts in QM (quantum mechanics), and is famous for the fact that it has never been observed, and specifically can’t be, even in principle. This is related to the equally famous ‘measurement problem’, whereby a quantum event becomes observable, and I would say, becomes ‘classical’, as in classical physics. My argument is that this is because Ψ only exists in the future of whoever (or whatever) is going to observe it (or interact with it). By expressing it specifically in those terms (of an observer), it doesn’t contradict relativity theory, quantum entanglement notwithstanding (another topic).
 
Some argue, like Carlo Rovelli (who knows a lot more about this topic than me), that Schrodinger’s equation and the concept of a wave function has led QM astray, arguing that if we’d just stuck with Heisenberg’s matrices, there wouldn’t have been a problem. Schrodinger himself demonstrated that his wave function approach and Heisenberg’s matrix approach are mathematically equivalent. And this is why we have so many ‘interpretations’ of QM, because they can’t be mathematically delineated. It’s the same with Feynman’s QED and Schwinger’s QFT, which Dyson showed were mathematically equivalent, along with Tomanaga’s approach, which got them all a Nobel prize, except Dyson.
 
As I pointed out on another post, physics is really just mathematical models of reality, and some are more accurate and valid than others. In fact, some have turned out to be completely wrong and misleading, like Ptolemy’s Earth-centric model of the solar system. So Rovelli could be right about the wave function. Speaking of reifying mathematical entities into physical reality, I had an online discussion with Qld Uni physicist, Mark John Fernee, who takes it a lot further than I do, claiming that 3 dimensional space (or 4 dimensional spacetime) is a mathematical abstraction. Yet, I think there really are 3 dimensions of space, because the number of dimensions affects the physics in ways that would be catastrophic in another hypothetical universe (refer John Barrow’s The Constants of Nature). So it’s more than an abstraction. This was a key point of difference I had with Fernee (you can read about it here).
 
All of this is really a preamble, because I think the most demonstrable and arguably most consequential example of the link between mathematics and reality is chaos theory, and it doesn’t involve reification. Having said that, this again led to a point of disagreement between myself and Fermee, but I’ll put that to one side for the moment, so as not to confuse you.
 
A lot of people don’t know that chaos theory started out as purely mathematical, largely due to one man, Henri Poincare. The thing about physical chaotic phenomena is that they are theoretically deterministic yet unpredictable simply because the initial conditions of a specific event can’t be ‘physically’ determined. Now some physicists will tell you that this is a physical limitation of our ability to ‘measure’ the initial conditions, and infer that if we could, it would be ‘problem solved’. Only it wouldn’t, because all chaotic phenomena have a ‘horizon’ beyond which it’s impossible to make accurate predictions, which is why weather predictions can’t go reliably beyond 10 days while being very accurate over a few. Sabine Hossenfelder explains this very well.
 
But here’s the thing: it’s built into the mathematics of chaos. It’s impossible to calculate the initial conditions because you need to do the calculation to infinite decimal places. Paul Davies gives an excellent description and demonstration in his book, The Cosmic Blueprint. (this was my point-of-contention with Fernee, talking about coin-tosses).
 
As I discussed on another post, infinity is a mathematical concept that appears to have little or no relevance to reality. Perhaps the Universe is infinite in space – it isn’t in time – but if it is, we might never know. Infinity avoids empirical confirmation almost by definition. But I think chaos theory is the exception that proves the rule. The reason we can’t determine the exact initial conditions of a chaotic event, is not just physical but mathematical. As Fernee and others have pointed out, you can manipulate a coin-toss to make it totally predictable, but that just means you’ve turned a chaotic event into a non-chaotic event (after all it’s a human-made phenomenon). But most chaotic events are natural, like the orbits of the planets and biological evolution. The creation of the Earth’s moon was almost certainly a chaotic event, without which complex life would almost certainly never have evolved, so they can be profoundly consequential as well as completely unpredictable.
 

Is randomness real or illusion?

 Let’s look at quantum mechanics (QM). I watched a YouTube video on Closer To Truth with Fred Alan Wolf, a theoretical physicist, whom I admit I’d never heard of. It’s worth watching the first 7 mins before he goes off on a speculative tangent that maybe dreams are a more fundamental level of reality, citing Australian Aboriginal ‘dreamtime’ mythology, of which I have some familiarity, though no scholarship.
 
In the first 7 mins he describes QM: its conceptual frustrations juxtaposed with its phenomenal successes. He gives a good synopsis, explaining how it describes a world we don’t actually experience, yet apparently underpins (my term, not his) the one we do. In particular, he explains:
 
There is a simple operation that takes you out of that space into (hits the table with his hand) this space. And that operation is simply multiplying what that stuff - that funny stuff – is, by itself (waves his hands in circles) in a time-reverse manner, called psi star psi (Ψ*Ψ) in the language of quantum physics.
 
What he’s describing is called the Born rule, which gives probabilities of finding that ‘stuff’ in the real world. By ‘real world’ I mean the one we are all familiar with and that he can hit his hand with. Ψ (pronounced sy) is of course the wave function in Schrodinger’s eponymous equation, and Schrodinger himself wrote a paper (in 1941) demonstrating that Born’s rule effectively multiplies the wave function by itself running backwards in time.
 
Now, some physicists argue that Ψ is just a convenient mathematical fiction and Carlo Rovelli went so far as to argue that it has led us astray (in one of his popular books). Personally, I think it describes the future, which explains why we never see it, or as soon as we try to, it disappears, and if we’re lucky, we get a particle or some other interaction, like a dot on a screen, all of which exist in our past. Note that everything we observe, including our own reflection in a mirror, exists in the past.
 
Wolf then goes on to speculate that the infinite possibilities we use for our calculations are perhaps the true reality. In his own words: What I’m interested in are the things we can’t see… And he makes an interesting point that most people don’t know: that if we don’t take into account the things we can’t see, ‘we get the wrong answers’.
 
And this is where it gets interesting, because he’s alluding to Feynman’s sum-over-histories methodology, which takes into account all the infinite paths that the particle (as wave function) can take. In fact, the more paths that are allowed for, the more accurate the calculation. Wolf doesn’t mention Feynman, but I’m sure that’s what he’s referring to.
 
Feynman’s key insight into QM was that it obeys the least-action principle, which is mathematically expressed as a Lagrangian. It’s the ‘least-action principle’ that determines where light goes through a change in medium (like glass), obeying Fermat’s law where it takes the path of ‘least time’. It also determines the path a ball follows if you throw it into the air by following the path of ‘maximum relativistic time’. I elaborate on this in another post.
 
There is something teleological about this principle, as if the ball, particle, light, ‘knows’ where it has to go. Freeman Dyson, who was a close collaborator with Feynman, argued that QM cannot describe the past, but only the future, and that only classical physics describes the past. So these infinitude of paths that are part of the calculation to determine the probability of where it will actually be ‘observed’ make more sense to me if they exist in the future. I don’t think we need a ‘dream state’ unless that’s a euphemism for the future.
 
Like Dyson, I don’t think we need consciousness to make a quantum phenomenon become real, but it does provide the reference point. In his own words:
 
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 thing about consciousness is that it exists in a ‘constant present’, as pointed out by Schrodinger himself (when he wasn’t talking about QM), so it logically correlates with 'a point of reference, to separate past from future', that Dyson refers to.
 
Schrodinger coined a term, ‘statistico-deterministic’, to describe quantum phenomena, because, at a statistical level, it can be very predictable, otherwise we wouldn’t be able to call it ‘successful’. He gives the example of radioactive decay (exploited in his eponymous cat thought experiment) whereby we can’t determine the decay of a single isotope, yet we can statistically determine the half-life of astronomical numbers of atoms very accurately, as everyone knows.
 
I contend that real randomness, that we all observe and are familiar with, is caused by chaos, but even this is a contentious idea. I like to give the example of tossing a coin, but a lot of physicists will tell you that tossing a coin is not random. In fact, I recently had a lengthy, but respectful, discourse with Mark John Fernee (physicist at Qld Uni) on Quora on this very topic. When I raised the specific issue of whether tossing a coin is ‘random’, he effectively argued that there are no random phenomena in physics. To quote him out of context:
 
Probability theory is built from statistical sampling. There is no assumed underlying physics.
 
The underlying physics can be deterministic, while a statistical distribution of events can indicate random behaviour. This is the assumption that is applied to every coin toss. Because this is just an assumption, you can cheat the system by using specific conditions that ensure deterministic outcomes.
 
What I am saying is that randomness is a statistical characterisation of outcomes that does not include any physical mechanism. As such, it is not a fundamental property of nature. (Emphasis in original)
 
I get the impression from what I’ve read that mathematicians have a different take on chaos to physicists, because they point out that you need to calculate initial conditions to infinite decimal places to achieve a 100% predicted outcome. Physicist, Paul Davies, provided a worked example in his 1988 book, The Cosmic Blueprint. I quoted Davies to Fernee during our ‘written’ conversation:
 
It is actually possible to prove that the activity of the jumping particle is every bit as random as tossing a coin.
 
The ‘jumping particle’ Davies referred to was an algorithm using clock arithmetic, that when graphed produced chaotic results, and he demonstrated that it would take a calculation to infinity to get it ‘exactly right’. Fernee was dismissive of this and gave it as an example of a popular science book leading laypeople (like myself) astray, which I thought was a bit harsh, as Davies actually goes into the mathematics in some detail, and I possibly misled Fernee by quoting just one sentence.
 
Just to be clear, Fernee doesn’t disagree that chaotic phenomena are impossible to predict; just that they are fully deterministic and, in his words, only ‘indicate random behaviour’.
 
Sabine Hossenfelder, who argues very strongly for superdeterminism, has a video demonstrating how predicting chaotic phenomena (like the weather) has a horizon (my term, not hers) of predictability that can never be exceeded, even in principle (10 days in the case of the weather).
 
So Fernee and Hossenfelder distinguish between what we ‘cannot know’ and what physically transpires. But my point is that chaotic phenomena, if rerun, will always produce a different result – it’s built into the mathematics underlying the activity – and includes significant life-changing phenomena like evolutionary biology and the orbits of the planets, as well as weather and earthquakes. Even the creation of the moon is believed to be a consequence of a chaotic event, without which life on Earth would never have evolved.
 
Note that both QM and chaos have mathematical underpinnings, and whilst most see that as modelling or a very convenient method of making predictions, I see it as more fundamental. I contend that mathematics transcends the Universe, yet it’s also a code that allows us to plumb Nature’s deepest secrets and fathom the dynamics of the Universe on all scales.

 

Follow-up (30 Mar 2024)

Following my discourse with Fernee, I reread Davies’ book, The Cosmic Blueprint (for the third time since I bought it in the late 80s), or at least the part that was relevant. I really did Davies a disservice by just quoting one sentence out of context. In fact, Davies goes to a lot of trouble to try and define what randomness means. He also acknowledges that, despite being totally unpredictable, chaotic phenomena are still ‘deterministic’ – it’s just the initial conditions that are unattainable (mathematically as well as physically). That is why, when you rerun a chaotic event, you get a different result, despite being so-called ‘deterministic’.
 
As well as the mathematical example I gave above, Davies discusses in detail 2 physical systems that are chaotic – the population of certain species of animals and the forcing of a pendulum (where a constant force is applied to a pendulum at a different frequency to its natural frequency). Marcus du Sautoy in his book, What We Cannot Know, interviews ex-pat Australian, Robert May (now a Member of the House of Lords), who did pioneering work on chaos theory in animal populations.
 
Davies quotes Ilya Prigogine concerning ‘…the conviction that the future is determined by the present…  We may perhaps even call it the founding myth of classical science.’
 
He also quotes Joseph Ford: ‘…the fact that determinism actually reigns only over a quite finite domain; outside this small haven of order lies a largely uncharted, vast wasteland of chaos where determinism has faded into an ephemeral memory of existence theorems and only randomness survives.’
 
And then Davies himself:
 
But in reality, our universe is not a linear Newtonian mechanical system; it is a chaotic system… No finite intelligence, however powerful, could anticipate what new forms or systems may come to exist in the future, The universe is in some sense open; it cannot be known what new levels of variety or complexity may be in store.
 
In light of these comments from last century, and considering that under Newton and Pascale, everyone thought that given enough information, the entire universe’s future could be foreseen, I see ‘strong determinism’ (as opposed to weak determinism) as a scientific ‘fashion’ that’s come back into favour. By ‘weak determinism’, I mean that all physical phenomena have a causal relationship; it’s just impossible to predict beyond a horizon, which is dependent on the nature of the phenomenon (whether it be the weather or the planets). Therefore, I think randomness is built into the Universe, and its principal mechanism is chaos, not quantum.

Consciousness, free will, determinism, chaos theory – all connected

 I’ve said many times that philosophy is all about argument. And if you’re serious about philosophy, you want to be challenged. And if you want to be challenged you should seek out people who are both smarter and more knowledgeable than you. And, in my case, Sabine Hossenfelder fits the bill.
 
When I read people like Sabine, and others whom I interact with on Quora, I’m aware of how limited my knowledge is. I don’t even have a university degree, though I’ve attempted a number of times. I’ve spent my whole life in the company of people smarter than me, including at school. Believe it or not, I still have occasional contact with them, through social media and school reunions. I grew up in a small rural town, where the people you went to school with feel like siblings.
 
Likewise, in my professional life, I have always encountered people cleverer than me – it provides perspective.
 
In her book, Existential Physics; A Scientist’s Guide to Life’s Biggest Questions, Sabine interviews people who are possibly even smarter than she is, and I sometimes found their conversations difficult to follow. To be fair to Sabine, she also sought out people who have different philosophical views to her, and also have the intellect to match her.
 
I’m telling you all this to put things in perspective. Sabine has her prejudices like everyone else, some of which she defends better than others. I concede that my views are probably more simplistic than hers, and I support my challenges with examples that are hopefully easy to follow. Our points of disagreement can be distilled down to a few pertinent topics, which are time, consciousness, free will and chaos. Not surprisingly, they are all related – what you believe about one, affects what you believe about the others.
 
Sabine is very strict about what constitutes a scientific theory. She argues that so-called theories like the multiverse have ‘no explanatory power’, because they can’t be verified or rejected by evidence, and she calls them ‘ascientific’. She’s critical of popularisers like Brian Cox who tell us that there could be an infinite number of ‘you(s)’ in an infinite multiverse. She distinguishes between beliefs and knowledge, which is a point I’ve made myself. Having said that, I’ve also argued that beliefs matter in science. She puts all interpretations of quantum mechanics (QM) in this category. She keeps emphasising that it doesn’t mean they are wrong, but they are ‘ascientific’. It’s part of the distinction that I make between philosophy and science, and why I perceive science as having a dialectical relationship with philosophy.
 
I’ll start with time, as Sabine does, because it affects everything else. In fact, the first chapter in her book is titled, Does The Past Still Exist? Basically, she argues for Einstein’s ‘block universe’ model of time, but it’s her conclusion that ‘now is an illusion’ that is probably the most contentious. This critique will cite a lot of her declarations, so I will start with her description of the block universe:
 
The idea that the past and future exist in the same way as the present is compatible with all we currently know.
 
This viewpoint arises from the fact that, according to relativity theory, simultaneity is completely observer-dependent. I’ve discussed this before, whereby I argue that for an observer who is moving relative to a source, or stationary relative to a moving source, like the observer who is standing on the platform of Einstein’s original thought experiment, while a train goes past, knows this because of the Doppler effect. In other words, an observer who doesn’t see a Doppler effect is in a privileged position, because they are in the same frame of reference as the source of the signal. This is why we know the Universe is expanding with respect to us, and why we can work out our movement with respect to the CMBR (cosmic microwave background radiation), hence to the overall universe (just think about that).
 
Sabine clinches her argument by drawing a spacetime diagram, where 2 independent observers moving away from each other, observe a pulsar with 2 different simultaneities. One, who is traveling towards the pulsar, sees the pulsar simultaneously with someone’s birth on Earth, while the one travelling away from the pulsar sees it simultaneously with the same person’s death. This is her slam-dunk argument that ‘now’ is an illusion, if it can produce such a dramatic contradiction.
 
However, I drew up my own spacetime diagram of the exact same scenario, where no one is travelling relative to anyone one else, yet create the same apparent contradiction.

 My diagram follows the convention in that the horizontal axis represents space (all 3 dimensions) and the vertical axis represents time. So the 4 dotted lines represent 4 observers who are ‘stationary’ but ‘travelling through time’ (vertically). As per convention, light and other signals are represented as diagonal lines of 45 degrees, as they are travelling through both space and time, and nothing can travel faster than them. So they also represent the ‘edge’ of their light cones.
 
So notice that observer A sees the birth of Albert when he sees the pulsar and observer B sees the death of Albert when he sees the pulsar, which is exactly the same as Sabine’s scenario, with no relativity theory required. Albert, by the way, for the sake of scalability, must have lived for thousands of years, so he might be a tree or a robot.
 
But I’ve also added 2 other observers, C and D, who see the pulsar before Albert is born and after Albert dies respectively. But, of course, there’s no contradiction, because it’s completely dependent on how far away they are from the sources of the signals (the pulsar and Earth).
 
This is Sabine’s perspective:
 
Once you agree that anything exists now elsewhere, even though you see it only later, you are forced to accept that everything in the universe exists now. (Her emphasis.)
 
I actually find this statement illogical. If you take it to its logical conclusion, then the Big Bang exists now and so does everything in the universe that’s yet to happen. If you look at the first quote I cited, she effectively argues that the past and future exist alongside the present.
 
One of the points she makes is that, for events with causal relationships, all observers see the events happening in the same sequence. The scenario where different observers see different sequences of events have no causal relationships. But this begs a question: what makes causal events exceptional? What’s more, this is fundamental, because the whole of physics is premised on the principle of causality. In addition, I fail to see how you can have causality without time. In fact, causality is governed by the constant speed of light – it’s literally what stops everything from happening at once.
 
Einstein also believed in the block universe, and like Sabine, he argued that, as a consequence, there is no free will. Sabine is adamant that both ‘now’ and ‘free will’ are illusions. She argues that the now we all experience is a consequence of memory. She quotes Carnap that our experience of ‘past, present and future can be described and explained by psychology’ – a point also made by Paul Davies. Basically, she argues that what separates our experience of now from the reality of no-now (my expression, not hers) is our memory.
 
Whereas, I think she has it back-to-front, because, as I’ve pointed out before, without memory, we wouldn’t know we are conscious. Our brains are effectively a storage device that allows us to have a continuity of self through time, otherwise we would not even be aware that we exist. Memory doesn’t create the sense of now; it records it just like a photograph does. The photograph is evidence that the present becomes the past as soon as it happens. And our thoughts become memories as soon as they happen, otherwise we wouldn’t know we think.
 
Sabine spends an entire chapter on free will, where she persistently iterates variations on the following mantra:
 
The future is fixed except for occasional quantum events that we cannot influence.
 
But she acknowledges that while the future is ‘fixed’, it’s not predictable. And this brings us to chaos theory. Sabine discusses chaos late in the book and not in relation to free will. She explicates what she calls the ‘real butterfly effect’.
 
The real butterfly effect… means that even arbitrarily precise initial data allow predictions for only a finite amount of time. A system with this behaviour would be deterministic and yet unpredictable.
 
Now, if deterministic means everything physically manifest has a causal relationship with something prior, then I agree with her. If she means that therefore ‘the future is fixed’, I’m not so sure, and I’ll explain why. By specifying ‘physically manifest’, I’m excluding thoughts and computer algorithms that can have an effect on something physical, whereas the cause is not so easily determined. For example, In the case of the algorithm, does it go back to the coder who wrote it?
 
My go-to example for chaos is tossing coins, because it’s so easy to demonstrate and it’s linked to probability theory, as well as being the very essence of a random event. One of the key, if not definitive, features of a chaotic phenomenon is that, if you were to rerun it, you’d get a different result, and that’s fundamental to probability theory – every coin toss is independent of any previous toss – they are causally independent. Unrepeatability is common among chaotic systems (like the weather). Even the Earth and Moon were created from a chaotic event.
 
I recently read another book called Quantum Physics Made Me Do It by Jeremie Harris, who argues that tossing a coin is not random – in fact, he’s very confident about it. He’s not alone. Mark John Fernee, a physicist with Qld Uni, in a personal exchange on Quora argued that, in principle, it should be possible to devise a robot to perform perfectly predictable tosses every time, like a tennis ball launcher. But, as another Quora contributor and physicist, Richard Muller, pointed out: it’s not dependent on the throw but the surface it lands on. Marcus du Sautoy makes the same point about throwing dice and provides evidence to support it.
 
Getting back to Sabine. She doesn’t discuss tossing coins, but she might think that the ‘imprecise initial data’ is the actual act of tossing, and after that the outcome is determined, even if can’t be predicted. However, the deterministic chain is broken as soon as it hits a surface.
 
Just before she gets to chaos theory, she talks about computability, with respect to Godel’s Theorem and a discussion she had with Roger Penrose (included in the book), where she says:
 
The current laws of nature are computable, except for that random element from quantum mechanics.
 
Now, I’m quoting this out of context, because she then argues that if they were uncomputable, they open the door to unpredictability.
 
My point is that the laws of nature are uncomputable because of chaos theory, and I cite Ian Stewart’s book, Does God Play Dice? In fact, Stewart even wonders if QM could be explained using chaos (I don’t think so). Chaos theory has mathematical roots, because not only are the ‘initial conditions’ of a chaotic event impossible to measure, they are impossible to compute – you have to calculate to infinite decimal places. And this is why I disagree with Sabine that the ‘future is fixed’.
 
It's impossible to discuss everything in a 223 page book on a blog post, but there is one other topic she raises where we disagree, and that’s the Mary’s Room thought experiment. As she explains it was proposed by philosopher, Frank Jackson, in 1982, but she also claims that he abandoned his own argument. After describing the experiment (refer this video, if you’re not familiar with it), she says:
 
The flaw in this argument is that it confuses knowledge about the perception of colour with the actual perception of it.
 
Whereas, I thought the scenario actually delineated the difference – that perception of colour is not the same as knowledge. A person who was severely colour-blind might never have experienced the colour red (the specified colour in the thought experiment) but they could be told what objects might be red. It’s well known that some animals are colour-blind compared to us and some animals specifically can’t discern red. Colour is totally a subjective experience. But I think the Mary’s room thought experiment distinguishes the difference between human perception and AI. An AI can be designed to delineate colours by wavelength, but it would not experience colour the way we do. I wrote a separate post on this.
 
Sabine gives the impression that she thinks consciousness is a non-issue. She talks about the brain like it’s a computer.
 
You feel you have free will, but… really, you’re running a sophisticated computation on your neural processor.
 
Now, many people, including most scientists, think that, because our brains are just like computers, then it’s only a matter of time before AI also shows signs of consciousness. Sabine doesn’t make this connection, even when she talks about AI. Nevertheless, she discusses one of the leading theories of neuroscience (IIT, Information Integration Theory), based on calculating the amount of information processed, which gives a number called phi (Φ). I came across this when I did an online course on consciousness through New Scientist, during COVID lockdown. According to the theory, this number provides a ‘measure of consciousness’, which suggests that it could also be used with AI, though Sabine doesn’t pursue that possibility.
 
Instead, Sabine cites an interview in New Scientist with Daniel Bor from the University of Cambridge: “Phi should decrease when you go to sleep or are sedated… but work in Bor’s laboratory has shown that it doesn’t.”
 
Sabine’s own view:
 
Personally, I am highly skeptical that any measure consisting of a single number will ever adequately represent something as complex as human consciousness.
 
Sabine discusses consciousness at length, especially following her interview with Penrose, and she gives one of the best arguments against panpsychism I’ve read. Her interview with Penrose, along with a discussion on Godel’s Theorem, which is another topic, discusses whether consciousness is computable or not. I don’t think it is and I don’t think it’s algorithmic.
 
She makes a very strong argument for reductionism: that the properties we observe of a system can be understood from studying the properties of its underlying parts. In other words, that emergent properties can be understood in terms of the properties that it emerges from. And this includes consciousness. I’m one of those who really thinks that consciousness is the exception. Thoughts can cause actions, which is known as ‘agency’.
 
I don’t claim to understand consciousness, but I’m not averse to the idea that it could exist outside the Universe – that it’s something we tap into. This is completely ascientific, to borrow from Sabine. As I said, our brains are storage devices and sometimes they let us down, and, without which, we wouldn’t even know we are conscious. I don’t believe in a soul. I think the continuity of the self is a function of memory – just read The Lost Mariner chapter in Oliver Sacks’ book, The Man Who Mistook His Wife For A Hat. It’s about a man suffering from retrograde amnesia, so his life is stuck in the past because he’s unable to create new memories.
 
At the end of her book, Sabine surprises us by talking about religion, and how she agrees with Stephen Jay Gould ‘that religion and science are two “nonoverlapping magisteria!”. She makes the point that a lot of scientists have religious beliefs but won’t discuss them in public because it’s taboo.
 
I don’t doubt that Sabine has answers to all my challenges.
 
There is one more thing: Sabine talks about an epiphany, following her introduction to physics in middle school, which started in frustration.
 
Wasn’t there some minimal set of equations, I wanted to know, from which all the rest could be derived?
 
When the principle of least action was introduced, it was a revelation: there was indeed a procedure to arrive at all these equations! Why hadn’t anybody told me?
 
The principle of least action is one concept common to both the general theory of relativity and quantum mechanics. It’s arguably the most fundamental principle in physics. And yes, I posted on that too.

 

What we observe and what is reality are distinct in physics

 I’ve been doing this blog for 15 years now, and in that time some of my ideas have changed or evolved, and, in some areas, my knowledge has increased. As I’ve said on Quora a few times, I read a lot of books by people who know a lot more than me, especially in physics.
 
There is a boundary between physics and philosophy, the shoreline of John Wheeler’s metaphorical ‘island of knowledge in the infinite sea of ignorance’. To quote: “As the island grows so does the shoreline of our ignorance.” And I think ignorance is the key word here, because it’s basically speculation, which means some of us are wrong, including me, most likely. As I’ve often said, ‘Only future generations can tell us how ignorant the current generation is’. I can say that with a lot of confidence, just by looking at the history of science.
 
If this blog has a purpose beyond promoting my own pet theories and prejudices, it is to make people think.
 
Recently, I’ve been pre-occupied with determinism and something called superdeterminism, which has become one of those pet prejudices among physicists in the belief that it’s the only conclusion one can draw from combining relativity theory, quantum mechanics, entanglement and Bell’s theorem. Sabine Hossenfelder is one such advocate, who went so far as to predict that one day all other physicists will agree with her. I elaborate on this below.
 
Mark John Fernee (physicist with Qld Uni), with whom I’ve had some correspondence, is one who disagrees with her. I believe that John Bell himself proposed that superdeterminism was possibly the only resolution to the quandaries posed by his theorem. There are two other videos worth watching, one by Elijah Lew-Smith and a 50min one by Brian Greene, who doesn’t discuss superdeterminism. Nevertheless, Greene’s video gives the best and easiest to understand description of Bell’s theorem and its profound implications for reality.
 
So what is super-determinism, and how is it distinct from common or garden determinism? Well, if you watch the two relevant videos, you get two different answers. According to Sabine, there is no difference and it’s not really to do with Bell’s theorem, but with the measurement problem in QM. She argues that it’s best explained by looking at the double-slit experiment. Interestingly, Richard Feynman argued that all the problems associated with QM can be analysed, if not understood, by studying the double-slit experiment.
 
Sabine wrote an academic paper on the ‘measurement problem’, co-authored with Jonte R. Hance from the University of Bristol, which I’ve read and is surprisingly free of equations (not completely) but uses the odd term I’m unfamiliar with. I expect I was given a link by Fernee which I’ve since lost (I really can’t remember), but I still have a copy. One of her points is that as long as we have unsolved problems in QM, there is always room for different philosophical interpretations, and she and Hance discuss the most well-known ones. This is slightly off-topic, but only slightly, because even superdeterminism and its apparent elimination of free will is a philosophical issue.
 
Sabine argues that it’s the measurement that creates superdeterminism in QM, which is why she uses the double-slit experiment to demonstrate it. It’s because the ‘measurement’ ‘collapses’ the wave function and ‘determines’ the outcome, that it must have been ‘deterministic’ all along. It’s just that we don’t know it until a measurement is made. At least, this is my understanding of her argument.
 
The video by Elijah Lew-Smith gives a different explanation, focusing solely on Bell’s theorem. I found that it also required more than one viewing, but he makes a couple of points, which I believe go to the heart of the matter. (Greene’s video gives an easier-to-follow description, despite its length).
 
We can’t talk about an objective reality independent of measurement.
(Which echoes Sabine’s salient point in her video.)
 
And this point: There really are instantaneous interactions; we just can’t access them.
 
This is known as ‘non-locality’, and Brian Greene provides the best exposition I’ve seen, and explains how it’s central to Bell’s theorem and to our understanding of reality.
 
On the other hand, Lew-Smith explains non-locality without placing it at the centre of the discussion.
 
If I can momentarily go back to Sabine’s key argument, I addressed this in a post I wrote a few years back. Basically, I argued that you can only know the path an electron or photon takes retrospectively, after the measurement or observation has been made. Prior to that, QM tells us it’s in a superposition of states and we only have probabilities of where it will land. Curiously, I referenced a video by Sabine in a footnote, where she makes this point in her conclusion:
 
You don’t need to know what happens in the future because the particle goes to all points anyway. Except…  It doesn’t. In reality, it goes to only one point. So maybe the reason we need the measurement postulate is because we don’t take this dependency on the future seriously enough.
 
And to me, that’s what this is all about: the measurement is in the future of the wave function, and the path it takes is in the past. This, of course, is what Freeman Dyson claims: that QM cannot describe the past, only the future.
 
And if you combine this perspective with Lew-Smith’s comment about objective reality NOT being independent of the measurement, then objective reality only exists in the past, while the wave function and all its superpositional states exist in the future.
 
So how does entanglement fit into this? Well, this is the second point I highlighted, which is that ‘there really are instantaneous reactions, which we can’t access’, which is ‘non-locality’. And this, as Schrodinger himself proclaimed, is what distinguishes QM from classical physics. In classical physics, ‘locality’ means there is a relativistic causal connection and in entanglement there is not, which is why Einstein called it ‘spooky action at a distance’.
 
Bell’s theorem effectively tells us that non-locality is real, supported by experiment many times over, but you can’t use it to transmit information faster-than-light, so relativity is not violated in practical terms. But it does ask questions about simultaneity, which is discussed in Lew-Smith’s video. He demonstrates graphically that different observers will observe a different sequence of measurement, so we have disagreement, even a contradiction about which ‘measurement’ collapsed the wave function. And this is leads to superdeterminism, because, if the outcome is predetermined, then the sequence of measurement doesn’t matter.
 
And this gets to the nub of the issue, because it ‘appears’ that ‘objective reality’ is observer dependent. Relativity theory always gives the result from a specific observer’s point of view and different observers in different frames of reference can epistemically disagree. Is there a frame of reference that is observer independent? I always like to go back to the twin paradox, because I believe it provides an answer. When the twins reunite, they disagree on how much time has passed, yet they agree on where they are in space-time. There is not absolute time, but there is absolute space-time.
 
Did you know we can deduce the velocity that Earth travels relative to absolute space-time, meaning the overall observable Universe? By measuring the Doppler shift of the CMBR (cosmic microwave background radiation) in all directions, it’s been calculated that we are travelling at 350km/s in the direction of Pisces (ref., Paul Davies, About Time; Einstein’s Unfinished Revolution, 1995). They should teach this in schools.
 
Given this context, is it possible that entanglement is a manifestation of objective simultaneity? Not according to Einstein, who argued that: ‘The past, present and future is only a stubbornly persistent illusion’; which is based on the ‘fact’ that simultaneity is observer dependent. But Einstein didn’t live to see Bell’s theorem experimentally verified. Richard Muller, a prize-winning physicist and author (also on Quora) was asked what question he’d ask Einstein if he could hypothetically meet him NOW. I haven’t got a direct copy, but essentially Muller said he’d ask Einstein if he now accepted a ‘super-luminal connection’, given experimental confirmation of Bell’s theorem. In other words, entanglement is like an exception to the rule, where relativity strictly doesn’t apply.
 
Sabine with her co-author, Jonte Hance, make a passing comment that the discussion really hasn’t progressed much since Bohr and Einstein a century ago, and I think they have a point.
 
Mark Fernee, whom I keep mentioning on the sidelines, does make a distinction between determinism and superdeterminism, where determinism simply means that everything is causally connected to something, even if it’s not predictable. Chaos being a case-in-point, which he describes thus:
 
Where this determinism breaks down is with chaotic systems, such as three body dynamics. Chaotic systems are so sensitive to the initial parameters that even a slight inaccuracy can result in wildly different predictions. That's why predicting the weather is so difficult.
Overall, complexity limits the ability to predict the future, even in a causal universe.
 
On the other hand, superdeterminism effectively means the end of free will, and, in his own words, ‘free will is a contentious issue, even among physicists’.
 
Fernee provided a link to another document by Sabine, where she created an online forum specifically to deal with less than knowledgeable people about their disillusioned ideas on physics – crackpots and cranks. It occurred to me that I might fall into this category, but it’s for others to judge. I’m constantly reminded of how little I really know, and that I’m only fiddling around the edges, or on the ‘shoreline of ignorance’, as Wheeler described it, where there are many others far more qualified than me.
 
I not-so-recently wrote a post where I challenged a specific scenario often cited by physicists, where two observers hypothetically ‘observe’ contradictory outcomes of an event on a distant astronomical body that is supposedly happening simultaneously with them.
 
As I said before, relativity is an observer-dependent theory, almost by definition, and we know it works just by using the GPS on our smart-phones. There are algorithms that make relativistic corrections to the signals coming from the satellites, otherwise the map on your phone would not match the reality of your actual location.
 
What I challenge is the application of relativity theory to an event that the observer can’t observe, even in principle. In fact, relativity theory rules out a physical observation of a purportedly simultaneous event. So I’m not surprised that we get contradictory results. The accepted view among physicists is that each observer ‘sees’ a different ontology (one in the future and one in the past), whereas I contend that there is an agreed ontology that becomes observable at a later time, when it’s in both observers’ past. (Brian Greene has another video demonstrating the ‘conventional’ view among physicists.)
 
Claudia de Rahm is Professor of Physics at Imperial College London, and earlier this year, she gave a talk titled, What We Don’t Know About Gravity, where she made the revelatory point
that Einstein’s GR (general theory of relativity) predicted its own limitations. Basically, if you apply QM probabilities to extreme curvature spacetime, you get answers over 100%, so nonsense. GR and QM are mathematically incompatible if we try to quantise gravity, though QFT (quantum field theory) ‘works fine on the manifold of spacetime’, according to expert, Viktor T Toth.
 
Given that relativity theory, as it is applied, is intrinsically observer dependent, I question if it can be (reliably) applied to events that have no causal relation to the observer (meaning outside the observer's light cone, both past and future). Which is why I challenge its application to events the observer can't observe (refer 2 paragraphs ago).

 

Addendum: I changed the title so it's more consistent with the contents of the post. The previous title was Ignorance and bliss; philosophy and science. Basically, the reason we have different interpretations of the same phenomenon is because physics can only tell us about what we observe, and what that means for reality is often debatable; superdeterminism being a case in point. Many philosophers and scientists talk about a ‘gap’ between theory and reality, whereas I claim the gap is between the observation and reality, a la Kant.

Does the "unreasonable effectiveness of Mathematics" suggest we are in a simulation?

 This was a question on Quora, and I provided 2 responses: one being a comment on someone else’s post (whom I follow); and the other being my own answer.

Some years ago, I wrote a post on this topic, but this is a different perspective, or 2 different perspectives. Also, in the last year, I saw a talk given by David Chalmers on the effects of virtual reality. He pointed out that when we’re in a virtual reality using a visor, we trick our brains into treating it as if it’s real. I don’t find this surprising, though I’ve never had the experience. As a sci-fi writer, I’ve imagined future theme parks that were completely, fully immersive simulations. But I don’t believe that provides an argument that we live in a simulation, for reasons I provide in my Quora responses, given below.

 

Comment:

 

Actually, we create a ‘simulacrum’ of the ‘observable’ world in our heads, which is different to what other species might have. For example, most birds have 300 degree vision, plus they see the world in slow motion compared to us.

 

And this simulacrum is so fantastic it actually ‘feels’ like it exists outside your head. How good is that? 

 

But here’s the thing: in all these cases (including other species) that simulacrum must have a certain degree of faithfulness or accuracy with ‘reality’, because we interact with it on a daily basis, and, guess what? It can kill you.

 

But there is a solipsist version of this, which happens when we dream, but it won’t kill you, as far as we can tell, because we usually wake up.

 

Maybe I should write this as a separate answer.

 

And I did:

 

One word answer: No.

 

But having said that, there are 2 parts to this question, the first part being the famous quote from the title of Eugene Wigner’s famous essay. But I prefer this quote from the essay itself, because it succinctly captures what the essay is all about.

 

It is difficult to avoid the impression that a miracle confronts us here… or the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them.

 

This should be read in conjunction with another famous quote; this time from Einstein:

 

The most incomprehensible thing about the Universe is that it’s comprehensible.

 

And it’s comprehensible because its laws can be rendered in the language of mathematics and humans have the unique ability (at least on Earth) to comprehend that language even though it appears to be neverending.

 

And this leads into the philosophical debate going as far back as Plato and Aristotle: is mathematics invented or discovered?

 

The answer to that question is dependent on how you look at mathematics. Cosmologist and Fellow of the Royal Society, John Barrow, wrote a very good book on this very topic, called Pi in the Sky. In it, he makes the pertinent point that mathematics is not so much about numbers as the relationships between numbers. He goes further and observes that once you make this leap of cognitive insight, a whole new world opens up.

 

But here’s the thing: we have invented a system of numbers, most commonly to base 10, (but other systems as well), along with specific operators and notations that provide a language to describe and mentally manipulate these relationships. But the relationships themselves are not created by us: they become manifest in our explorations. To give an extremely basic example: prime numbers. You cannot create a prime number, they simply exist, and you can’t change one into a non-prime number or vice versa. And this is very basic, because primes are called the atoms of mathematics, because all the other ‘natural’ numbers can be derived from them.

 

An interest in the stars started early among humans, and eventually some very bright people, mainly Kepler and Newton, came to realise that the movement of the planets could be described very precisely by mathematics. And then Einstein, using Riemann geometry, vectors, calculus and matrices and something called the Lorenz transformation, was able to describe the planets even more accurately and even provide very accurate models of the entire observable universe, though recently we’ve come to the limits of this and we now need new theories and possibly new mathematics.

But there is something else that Einstein’s theories don’t tell us and that is that the planetary orbits are chaotic, which means they are unpredictable and that means eventually they could actually unravel. But here’s another thing: to calculate chaotic phenomena requires a computation to infinite decimal places. Therefore I contend the Universe can’t be a computer simulation. So that’s the long version of NO.

 

 

Footnote: Both my comment and my answer were ‘upvoted’ by Eric Platt, who has a PhD in mathematics (from University of Houston) and was a former software engineer at UCAR (University Corporation for Atmospheric Research).

Is infinity real?

 In some respects, I think infinity is what delineates mathematics from the ‘Real’ world, meaning the world we can all see and touch and otherwise ‘sense’ through an ever-expanding collection of instruments. To give an obvious example, calculus is used extensively in engineering and physics to determine physical parameters to great accuracy, yet the method requires the abstraction of infinitesimals at its foundation.

Sabine Hossenfelder, whom I’ve cited before, provides a good argument that infinity doesn’t exist in the real world, and Norman Wildberger even argues it doesn’t exist in mathematics because, according to his worldview, mathematics is defined only by what is computable. I won’t elaborate on his arguments but you can find them on YouTube.

 

I was prompted to write about this after reading the cover feature article in last week’s New Scientist by Timothy Revell, who is New Scientist’s deputy US editor. The article was effectively a discussion about the ‘continuum hypothesis’, which, following its conjecture by Georg Cantor, is still in the ‘undecidable’ category (proved neither true nor false). Basically, there are countable infinities and uncountable infinities, which was proven by Cantor and is uncontentious (with the exception of mathematical fringe-dwellers like Wildberger). The continuum hypothesis effectively says that there is no category of infinity in between, which I won’t go into because I don’t know enough about it. 

 

But I do understand Cantor’s arguments that demonstrate how the rational numbers are ‘countably infinite’ and how the Real numbers are not. To appreciate the extent of the mathematical universe (in numbers) to date, I recommend this video by Matt Parker. Sabine Hossenfelder, whom I’ve already referenced, gives a very good exposition on countable and uncountable infinities in the video linked above. She also explains how infinities are dealt with in physics, particularly in quantum mechanics, where they effectively cancel each other out. 

 

Sabine argues that ‘reality’ can only be determined by what can be ‘measured’, which axiomatically rules out infinity. She even acknowledges that the Universe could be physically infinite, but we wouldn’t know. Marcus du Sautoy, in his book, What We Cannot Know, argues that it might remain forever unknowable, if that’s the case. 

 

Nevertheless, Sabine argues that infinity is ‘real’ in mathematics, and I would agree. She points out that infinity is a concept that we encounter early, because it’s implicit in our counting numbers. No matter how big a number is, there is always a bigger one. Infinities are intrinsic to many of the unsolved problems in mathematics, and not just Cantor’s continuum hypothesis. There are 3 involving primes that are well known: the Goldbach conjecture, the twin prime conjecture and Riemann’s hypothesis, which is the most famous unsolved problem in mathematics, at the time of writing. In all these cases, it’s unknown if they’re true to infinity.

 

Without getting too far off the track, the Riemann hypothesis contends that all the non-trivial zeros of the Riemann Zeta function lie on a line in the complex plane which is 1/2 + ib. In other words, all the (nontrivial) zeros have Real part 1/2. The thing is that we already know there are an infinite number of them, we just don’t know if there are any that break that rule. The curious thing about infinities is that we are relatively comfortable with them, even though we can’t relate to them in the physical world, and they can never be computed. As I said in my opening paragraph, it’s what separates mathematics from reality.

 

And this leads one to consider what mathematics is, if it’s not reality. Not so recently, I had a discussion with someone on Quora who argued that mathematics is ‘fiction’. Specifically, they argued that any mathematics with no role in the physical universe is fiction. There is an immediate problem with this perspective, because we often don’t find a role in the ‘real world’ for mathematical discoveries, until decades, or even centuries later.

 

I’ve argued in another post that there is a fundamental difference between a physics equation and a purely mathematical equation that many people are not aware of. Basically, physics equations, like Einstein’s most famous, E = mc², have no meaning outside the physical universe; they deal with physical parameters like mass, energy, time and space.

 

On the other hand, there are mathematical relationships like Euler’s famous identity, e^iπ + 1 = 0, which has no meaning in the physical world, unless you represent it graphically, where it is a point on a circle in the complex plane. Talking about infinity, π famously has an infinite number of digits, and Euler’s equation, from which the identity is derived, comes from the sum of two infinite power series.

 

And this is why many mathematicians and physicists treat mathematics as a realm that already exists independently of us, known as mathematical Platonism. John Barrow made this point in his excellent book, Pi in the Sky, where he acknowledges it has quasi-religious connotations. Paul Davies invokes an imaginative metaphor of there being a ‘mathematical warehouse’ where ‘Mother Nature’, or God (if you like), selects the mathematical relationships which make up the ‘laws of the Universe’. And this is the curious thing about mathematics: that it’s ‘unreasonably effective in describing the natural world’, which Eugene Wigner wrote an entire essay on in the 1960s.

 

Marcus du Sautoy, whom I’ve already mentioned, points out that infinity is associated with God, and both he and John Barrow have suggested that the traditional view of God could be replaced with mathematics. Epistemologically, I think mathematics has effectively replaced religion in describing both the origins of the Universe and its more extreme phenomena. 

 

If one looks at the video I cited by Matt Parker, it’s readily apparent that there is infinitely more mathematics that we don’t know compared to what we do know, and Gregory Chaitin has demonstrated that there are infinitely more incomputable Real numbers than computable Reals. This is consistent with Godel’s famous Incompleteness Theorem that counter-intuitively revealed that there is a mathematical distinction between ‘proof’ and ‘truth’. In other words, in any consistent, axiom-based system of mathematics there will always exist mathematical truths that can’t be proved within that system, which means we need to keep expanding the axioms to determine said truths. This implies that mathematics is a never-ending epistemological endeavour. And, if our knowledge of the physical world is dependent on our knowledge of mathematics, then it’s arguably a never-ending endeavour as well.

 

I cannot leave this topic without discussing the one area where infinity and the natural world seem to intersect, which literally has world-changing consequences. I’m talking about chaos theory, which is dependent on the sensitivity of initial conditions. Paul Davies, in his book, The Cosmic Blueprint, actually provides an example where he shows that, mathematically, you have to calculate the initial conditions to infinite decimal places to make a precise prediction. Sabine Hossenfelder has a video on chaos where she demonstrates how it’s impossible to predict the future of a chaotic event beyond a specific horizon. This horizon varies – for the weather it’s around 10 days and for the planetary orbits it’s 10s of millions of years. Despite this, Sabine argues that the Universe is deterministic, which I’ve discussed in another post.

 

Mark John Fernee (physicist with Queensland University and regular Quora contributor) also argues that the universe is deterministic and that chaotic events are unpredictable because we can’t measure the initial conditions accurately enough. He’s not alone among physicists, but I believe it’s in the mathematics.

 

I point to coin tossing, which is the most common and easily created example of chaos. Marcus du Sautoy uses the tossing of dice, which he discusses in his aforementioned book, and in this video. The thing about chaotic events is that if you were to rerun them, you’d get a different result and that goes for the whole universe. Tossing coins is also associated with probability theory, where the result of any individual toss is independent of any previous toss with the same coin. That could only be true if chaotic events weren’t repeatable.

 

There is even something called quantum chaos, which I don’t know a lot about, but it may have a connection to Riemann’s hypothesis (mentioned above). Certainly, Riemann’s hypothesis is linked to quantum mechanics via Hermitian matrices, supported by relevant data (John Derbyshire, Prime Obsession). So, mathematics is related to the natural world in ever-more subtle and unexpected ways.

 

Chaos drives the evolvement of the Universe on multiple scales, including biological evolution and the orbits of planets. If chaos determines our fates, then infinities may well play the ultimate role.

 

 

Addendum: I made a very simple yet unforgivable mistake (since corrected), whereby I said the Zeta zeros in Riemann's Hypothesis were of the form a + 1/2ib, when it's the other way around: 1/2 + ib. So apologies.

Is the Universe deterministic?

 I’ve argued previously, and consistently, that the Universe is not deterministic; however, many if not most physicists believe it is. I’ve even been critical of Einstein for arguing that the Universe is deterministic (as per his famous dice-playing-God statement). 

Recently I’ve been watching YouTube videos by theoretical physicist, Sabine Hossenfelder, and I think she’s very good and I highly recommend her. Hossenfelder is quite adamant that the Universe is deterministic, and her video arguing against free will is very compelling and thought-provoking. I say this, because she addresses all the arguments I’ve raised in favour of free will, plus she has supplementary videos to support her arguments.

 

In fact, Hossenfelder states quite unequivocally towards the end of the video that ‘free will is an illusion’ and, in her own words, ‘needs to go into the rubbish bin’. Her principal argument, which she states right at the start, is that it’s ‘incompatible with the laws of nature’. She contends that the Universe is completely deterministic right from the Big Bang. She argues that everything can be described by differential equations, including gravity and quantum mechanics (QM), which she expounds upon in some detail in another video. 

 

My immediate reaction to this: is what about Poincare and chaos theory? Don’t worry, she addresses that as well. In fact, she has a couple of videos on chaos theory (though one is really about weather and climate change), which I’d recommend.

 

The standard definition of chaos is that it’s deterministic but unpredictable, which seems to be an oxymoron. As she points out, chaotic phenomena (which includes the weather and the orbits of the planet, among many other things, like evolution) are dependent on the ‘initial conditions’. An infinitesimal change in the initial conditions will result in a different outcome. The word ‘infinitesimal’ is the key here, because you need to work out the initial conditions to an infinite decimal place to get the answer. That’s why it’s not predictable. As to whether it’s deterministic, I think that’s another matter.

 

To overcome this apparent paradox, I prefer to say it’s indeterminable, which is not contentious. Hossenfelder explains, using a subtly different method, that you can mathematically prove, for any chaotic system, that you can only forecast to a finite time in the future, no matter how detailed your calculation (it’s worth watching her video, just to see this).

 

Because the above definition for chaos seems to lead to a contradiction or, at best, an oxymoron, I prefer another definition that is more pragmatic and is mostly testable (though not always). Basically, if you rerun a chaotic phenomenon, you’ll get a different outcome. The best known example is tossing a coin. It’s well known in probability theory (in fact it’s an axiom) that the result of the next coin toss is independent of all coin tosses that may have gone before. The reason for this is that coin tosses are chaotic. The same principle applies to throwing dice, and Marcus du Sautoy expounds on the chaos of throwing dice in this video. So, tossing coins and throwing dice are considered ‘random’ events in probability theory, but Hossenfelder contends they are totally deterministic; just unpredictable.

 

Basically, she’s arguing that just because we can’t calculate the initial conditions, they still happened and therefore everything that arises from them is deterministic. Du Sautoy (whom I referenced above) in the same video and in his book, What We Cannot Know, cites physicist turned theologian, John Polkinghorne, that chaos provides the perfect opportunity for an interventionist God – a point I’ve made myself (though I’m not arguing for an interventionist God). I’m currently reading Troy by Stephen Fry, an erudite rendition based on Homer’s tale, and it revolves around the premise that one’s destiny is largely predetermined by the Gods. The Hindu epic, Mahabharata, also portrays the notion of destiny that can’t be avoided. Leonard Cohen once remarked upon this in an interview, concerning his song, If It Be Your Will. In fact, I contend that you can’t believe in religious prophecy if you don’t believe in a deterministic universe. My non-belief in a deterministic universe is the basis of my argument against prophecy. And my argument against determinism is based on chaos and QM (which I’ll come to shortly).

 

Of course, one can’t turn back the clock and rerun the Universe, and, as best I can tell, that’s Hossenfelder’s sole argument for a deterministic universe – it can’t be changed and it can’t be predicted. She mentions Laplace’s Demon, who could hypothetically calculate the future of every particle in the Universe. But Laplace’s Demon is no different to the Gods of prophecy – it can do the infinite calculation that we mortals can’t do.

 

I have to concede that Hossenfelder could be right, based on the idea that the initial conditions obviously exist and we can’t rewind the clock to rerun the Universe. However, tossing coins and throwing dice demonstrate unequivocally that chaotic phenomena only become ‘known’ after the event and give different outcomes when rerun. 

 

So, on that basis, I contend that the future is open and unknowable and indeterminable, which leads me to say, it’s also non-deterministic. It’s a philosophical position based on what I know, but so is Hossenfelder’s, even though she claims otherwise: that her position is not philosophical but scientific.

 

Of course, Hossenfelder also brings up QM, and explains it is truly random but it’s also time reversible, which can be demonstrated with Schrodinger’s equation. She makes the valid point that the inherent randomness in QM doesn’t save free will. In fact, she says, ‘everything is either determined or random, neither of which are affected by free will’. However, she makes the claim that all the particles in our brain are quantum mechanically time reversible and therefore deterministic. However, I contend that the wave function that allows this time reversibility only exists in the future, which is why it’s never observed (I acknowledge that’s a personal prejudice). On the other hand, many physicists contend that the wave function is a purely mathematical construct that has no basis in reality.

 

My argument is that it’s only when the wave function ‘collapses’ or ‘decoheres’ that a ‘real’ physical event is observed, which becomes classical physics. Freeman Dyson argued something similar. Like chaotic events, if you were to rerun a quantum phenomenon you’d get a different outcome, which is why one can only deal in probabilities until an ‘observation’ is made. Erwin Schrodinger coined the term ‘statistico-deterministic’ to describe QM, because at a statistical level, quantum phenomena are predictable. He gives the example of radioactive decay, which we can predict holistically very accurately with ‘half-lives’, but you can’t predict the decay of an individual isotope at all. I argue that, both in the case of QM and chaos, you have time asymmetry, which means that if you could hypothetically rewind the clock before the wave function collapse or some initial conditions (whichever the case), you would witness a different outcome.

 

Hossenfelder sums up her entire thesis with the following statement:

 

...how ever you want to define the word [free will], we still cannot select among several possible different futures. This idea makes absolutely no sense if you know anything about physics.

 

Well, I know enough about physics to challenge her inference that there are no ‘possible different futures’. Hossenfelder, herself, knows that alternative futures are built-into QM, which is why the multiple worlds interpretation is so popular. And some adherents of the Copenhagen interpretation claim that you do get to ‘choose’ (though I don’t). If the wave function describes the future, it can have a multitude of future paths, only one of which becomes reality in the past. This derives logically from Dyson’s interpretation of QED.

 

Of course, none of this provides an argument for free will, even if the Universe is not deterministic.

 

Hossenfelder argues that the brain’s software (her term) runs calculations that determine our decisions, while giving the delusion of free will. I thought this was her best argument:

 

Your brain is running a calculation, and while it is going on you do not know the outcome of that calculation. So the impression of free will comes from our ‘awareness’ that we think about what we do, along with our inability to predict the result of what we are thinking.

 

You cannot separate the idea of free will from the experience of consciousness. In another video, Hossenfelder expresses scepticism at all the mathematical attempts to describe or explain consciousness. I’ve argued previously that if we didn’t all experience consciousness, science would tell us that it is an illusion just like free will is. That’s because science can’t explain the experience of consciousness any better than it can explain the intuitive sense of free will that most of us take for granted.

 

Leaving aside the use of the words, ‘calculation’ and ‘software’, which allude to the human brain being a computer, she’s right that much of our thinking occurs subconsciously. All artists are aware of this. As a storyteller, I know that the characters and their interactions I render on the page (or on a computer screen) largely come from my subconscious. But everyone experiences this in dreams. Do you think you have free will in a dream? In a so-called ‘lucid dream’, I’d say, yes.

 

I would like to drop the term, free will, along with all its pseudo-ontological baggage, and adopt another term, ‘agency’. Because it’s agency that we all believe we have, wherever it springs from. We all like to believe we can change our situation or exert some control over it, and I’d call that agency. And it requires a conscious effort – an ability to turn a thought into an action. In fact, I’d say it’s a psychological necessity: without a sense of agency, we might as well be automatons.

 

I will finish with an account of free will in extremis, as told by London bomber survivor, Gill Hicks. Gill Hicks was only one person removed from the bomber in one of the buses, and she lost both her legs. As she tells it, she heard a voice, like we do in a dream, and it was a female voice and it was ‘Death’ and it beckoned to her and it was very inviting; it was not tinged with fear at all. And then she heard another voice, which was male and it was ‘Life’, and it told her that if she chose to live she had a destiny to fulfil. So she had a choice, which is exactly how we define free will and she consciously chose Life. As it turned out, she lost 70% of her blood and she had a hole in the back of her head from a set of keys. In the ambulance, she later learned that she was showing no signs of life – no pulse and she had flatlined – yet she was talking. The ambo told the driver, ‘Dead but talking.’ It was only because she was talking that he continued to attempt to save her life.

 

Now, I’m often sceptical about accounts of ‘near-death experiences’, because they often come across as contrived and preachy. But Gill Hicks comes across as very authentic; down-to-Earth, as we say in Oz. So I believe that what she recalled is what she experienced. I tell her story, because it represents exactly what Hossenfelder claims about free will: it defies a scientific explanation.

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.