Have we forgotten what ‘mind’ means?

 There is an obvious rejoinder to this, which is, did we ever know what ‘mind’ means? Maybe that’s the real question I wanted to ask, but I think it’s better if it comes from you. The thing is that we have always thought that ‘mind’ means something, but now we are tending to think, because we have no idea where it comes from, that it has no meaning at all. In other words, if it can’t be explained by science, it has no meaning. And from that perspective, the question is perfectly valid.
 
I’ve been watching a number of videos hosted by Curt Jaimungal, whom I assume has a physics background. For a start, he’s posted a number of video interviews with a ‘Harvard scientist’ on quantum mechanics, and he provided a link (to me) of an almost 2hr video he did with Sabine Hossenfelder, and they talked like they were old friends. I found it very stimulating and I left a fairly long comment that probably no one will read.
 
Totally off-topic, but Sabine’s written a paper proposing a thought-experiment that would effectively test if QM and GR (gravity) are compatible at higher energies. She calculated the energy range and if there is no difference to the low energy experiments already conducted, it effectively rules out a quantum field for gravity (assuming I understand her correctly). I expressed my enthusiasm for a real version to be carried out, and my personal, totally unfounded prediction that it would be negative (there would be no difference).
 
But there are 2 videos that are relevant to this topic and they both involve Stephen Wolfram (who invented Mathematica). I’ve referenced him in previous posts, but always second-hand, so it was good to hear him first-hand. In another video, also hosted by Jaimungal, Wolfram has an exchange with Donald Hoffman, whom I’ve been very critical of in the past. But the truth is that all of these people know much more about their fields than me. I’ll get to the exchange with Hoffman later.
 
I have the impression from Gregory Chaitin, in particular, that Wolfram argues that the Universe is computable; a philosophical position I’ve argued against, mainly because of chaos theory. I’ve never known Wolfram to mention chaos theory, and he certainly doesn’t in the 2 videos I reference here, and I’ve watched them a few times.
 
Jaimungal introduces the first video (with Wolfram alone) by asking him about his ‘observer theory’ and ‘what if he’s right about the discreteness of space-time’ and ‘computation underlying the fundament?’ I think it’s this last point which goes to the heart of their discussion. Wolfram introduces a term called the Ruliad, which I had to look up. I came across 2 definitions, both of which seem relevant to the discussion.
 
A concept that describes all possible computations and rule-based systems, including our physical universe, mathematics, and everything we experience.
 
A meta-structural domain that encompasses every possible rule-based system, or computational eventuality, that can describe any universe or mathematical structure.
 
Wolfram confused me when he talked about ‘computational irreducibility’, which infers that there are some things that are not computable, to which I agree. But then later he seemed to argue that everything we can know is computable, and things we don’t know now are only unknowable because we’re yet to find their computable foundation. He argues that there are ‘slices of reducible computability’ within the ‘computational irreducibility’, which is how we do mathematical physics.
 
Towards the end of the video, he talks specifically about biology, saying, ‘there is no grand theory of biology’, like we attempt in physics. He has a point. I’ve long argued that natural selection is not the whole story, and there is a mystery inherent in DNA, in as much as it’s a code whose origin and evolvement is still unknown. Paul Davies attempted to tackle this in his book, The Demon in the Machine, because it’s analogous to software code and it’s information based. This means that it could, in principle, be mathematical, which means it could lead to a biological ‘theory of everything’, which I assume is what Wolfram is claiming is lacking.
 
However, I’m getting off-track again. At the start of the video, Wolfram specifically references the Copernican revolution, because it was not just a mathematical reformulation, but it changed our entire perspective of the Universe (we are not at the centre) without changing how we experience it (we are standing still, with the sky rotating around us). At the end of the day, we have mathematical models, and some are more accurate than others, and they all have limitations – there is no all-encompassing mathematical TOE (Theory of Everything). There is no Ruliad, as per the above definitions, and Wolfram acknowledges that while apparently arguing that everything is computable.
 
I find it necessary to bring Kant into this, and his concept of the ‘thing-in-itself’ which we may never know, but only have a perception of. My argument, which I’ve never seen anyone else employ, is that mathematics is one of our instruments of perception, just like our telescopes and particle accelerators and now, our gravitational wave detectors. Our mathematical models, be they GR (general relativity), QFT or String Theory, are perceptual and conceptual tools, whose veracity are ultimately determined by empirical evidence, which means they can only be applied to things that can be measured. And I think this leads to an unstated principle that if something can’t be measured it doesn’t exist. I would put ‘mind’ in that category.
 
And this allows me to segue into the second video, involving Donald Hoffman, because he seems to argue that mind is all that there is, and it has a mathematical foundation. He put forward his argument (which I wrote about recently) that, using Markovian matrices, he’s developed probabilities that apparently predict ‘qualia’, which some argue are the fundaments of consciousness. Wolfram, unlike the rest of us, actually knows what Hoffman is talking about and immediately had a problem that his ‘mathematical model’ led to probabilities and not direct concrete predictions. Wolfram seemed to argue that it breaks the predictive chain (my terminology), but I confess I struggled to follow his argument. I would have liked to ask: what happens with QM, which can only give us probabilities? In that case, the probabilities, generated by the Born Rule, are the only link between QM and classical physics – a point made by Mark John Fernee, among others.
 
But going back to my argument invoking Kant, it’s a mathematical model and not necessarily the thing-in-itself. There is an irony here, because Kant argued that space and time are a priori in the mind, so a projection, which, as I understand it, lies at the centre of Hoffman’s entire thesis. Hoffman argues that ‘spacetime is doomed’ since Nima Arkani-Hamed and his work on amplituhedrons, because (to quote Arkani-Hamed): This is a concrete example of a way in which the physics we normally associate with space-time and quantum mechanics arises from something more basic. In other words, Arkani-Hamed has found a mathematical substructure or foundation to spacetime itself, and Hoffman claims that he’s found a way to link that same mathematical substructure to consciousness, via Markovian matrices and his probabilities.
 
Hoffman analogises spacetime to wearing a VR headset and objects in spacetime to icons on a computer desktop, which seems to infer that the Universe is a simulation, though he’s never specifically argued that. I won’t reiterate my objections to Hoffman’s fundamental idealism philosophy, but if you have a mathematical model, however it’s formulated, its veracity can only be determined empirically, meaning we need to measure something. So, what is he going to measure? Is it qualia? Is it what people report what they think?
 
No. According to Hoffman, they can do empirical tests on spacetime (so not consciousness per se) that will determine if his mathematical model of consciousness is correct, which seems a very roundabout way of doing things. From what I can gather, he’s using a mathematical model of consciousness that’s already been developed (independently) to underpin reality, and then testing it on reality, thereby implying that consciousness is an intermediate step between the mathematical model and the reality. His ambition is to demonstrate that there is a causal relationship between consciousness and reality, when most argue that it’s the other way around. I return to this point below, with Wolfram’s response.
 
Wolfram starts off in his interaction with Hoffman by defining the subjective experience of consciousness that Hoffman has mathematically modelled and asking, can he apply that to an LLM (like ChatGPT, though he doesn’t specify) and therefore show that an LLM must be conscious? Wolfram argues that such a demonstration would categorically determine the ‘success’ (his term) of Hoffman’s theory, and Hoffman agreed.
 
I won’t go into detail (watch the video) but Hoffman concludes, quite emphatically, that ‘It’s not logically possible to start with non-conscious entities and have conscious agents emerge’ (my emphasis, obviously). Wolfram immediately responded (very good-naturedly), ‘That’s not my intuition’. He then goes on to say how that’s a Leibnizian approach, which he rejected back in the 1980s. I gather that it was around that time that Wolfram adopted and solidified (for want of a better word) his philosophical position that everything is ultimately computable. So they both see mathematics as part of the ‘solution’, but in different ways and with different conclusions.
 
To return to the point I raised in my introduction, Wolfram starts off in the first video (without Hoffman), that we have adopted a position that if something can’t be explained by science, then there is no other explanation – we axiomatically rule everything else out - and he seems to argue that this is a mistake. But then he adopts a position which is the exact opposite: that everything is “computational all the way down”, including concepts like free will. He argues: “If we can accept that everything is computational all the way down, we can stop searching for that.” And by ‘that’ he means all other explanations like mysticism or QM or whatever.
 
My own position is that mathematics, consciousness and physical reality form a triumvirate similar to Roger Penrose’s view. There is an interconnection, but I’m unsure if there is a hierarchy. I’ve argued that mathematics can transcend the Universe, which is known as mathematical Platonism, a view held by many mathematicians and physicists, which I’ve written about before.
 
I’m not averse to the view that consciousness may also exist beyond the physical universe, but it’s not something that can be observed (by definition). So far, I’ve attempted to discuss ‘mind’ in a scientific context, referencing 2 scientists with different points of view, though they both emphasise the role of mathematics in positing their views.
 
Before science attempted to analyse and put mind into an ontological box, we knew it as a purely subjective experience. But we also knew that it exists in others and even other creatures. And it’s the last point that actually triggered me to write this post and not the ruminations of Wolfram and Hoffman. When I interact with another animal, I’m conscious that it has a mind, and I believe that’s what we’ve lost. If there is a collective consciousness arising from planet Earth, it’s not just humans. This is something that I’m acutely aware of, and it has even affected my fiction.
 
The thing about mind is that it stimulates empathy, and I think that’s the key to the long-term survival of, not just humanity, but the entire ecosystem we inhabit. Is there a mind beyond the Universe? We don’t know, but I would like to think there is. In another recent post, I alluded to the Hindu concept of Brahman, which appealed to Erwin Schrodinger. You’d be surprised how many famous physicists were attracted to the mystical. I can think of Pauli, Einstein, Bohr, Oppenheimer – they all thought outside the box, as we like to say.
 
Physicists have no problem mentally conceiving 6 or more dimensions in String Theory that are ‘curled up’ so miniscule we can’t observe them. But there is also the possibility that there is a dimension beyond the universe that we can’t see. Anyone familiar with Flatland by Edwin Abbott (a story about social strata as much as dimensions), would know it expounds on our inherent inability to interact with higher dimensions. It’s occurred to me that consciousness may exist in another dimension, and we might ‘feel’ it occasionally when we interact with people who have died. I have experienced this, though it proves nothing. I’m a creative and a neurotic, so such testimony can be taken with a grain of salt.
 
I’ve gone completely off-track, but I think that both Wolfram and Hoffman may be missing the point, when, like many scientists, they are attempting to incorporate the subjective experience of mind into a scientific framework. Maybe it just doesn’t fit.

Mathematics, consciousness, reality

 I wish to emphasise the importance of following and listening to people you disagree with. (I might write another post on the pitfalls of ‘echo-chambers’ in social media, from which I’m not immune.)
 
I’ve been following Donald Hoffman ever since I reviewed an academic paper he wrote with Chetan Prakash called Objects of Consciousness, back in November 2016, though the paper was written in 2014 (so over 10 years ago). Back then, I have to admit, I found it hard to take him seriously, especially his views on evolution, and his go-to metaphor that objective reality was analogous to desktop icons on a computer.
 
His argument is similar to the idea that we live in a computer simulation, though he’s never said that, and I don’t think he believes we do. Nevertheless, he has compared reality to wearing a VR headset, which is definitely analogous to being in a computer simulation. As I have pointed out on other posts, I contend that we do create a model of reality in our ‘heads’, which is so ‘realistic’ that we all think it is reality. The thing is that our very lives depend on it being a very accurate ‘model’, so we can interact with the external reality that does exist outside our heads. This is one of my strongest arguments against Hoffman – reality can kill you, but simulations, including the ones we have when we sleep, which we call dreams, cannot.
 
So I’ve been following Hoffman, at least on YouTube, in the 8 years since I wrote that first critique. I read an article he wrote in New Scientist on evolution (can’t remember the date), which prompted me to write a letter-to-the-Editor, which was published. And whenever I come across him on YouTube: be it in an interview, a panel discussion or straight-to-video; I always watch and listen to what he has to say. What I’ve noticed is that he’s sharpened his scalpel, if I can use that metaphor, and that he’s changed his tack, if not his philosophical position. Which brings me to the reason for writing this post.
 
A year or two ago, I wrote a comment on one of his standalone videos, challenging what he said, and it was subsequently deleted, which is his prerogative. While I was critical, I don’t think I was particularly hostile – the tone was similar to a comment I wrote today on the video that prompted this discussion (see below).
 
Hoffman’s change of tack is not to talk about evolution at all, but spacetime and how it’s no longer ‘fundamental’. This allows him to argue that ‘consciousness’ is more fundamental than spacetime, via the medium of mathematics. And that’s effectively the argument he uses in this video, which, for brevity, I’ve distilled into one succinct sentence.
 
My approach, well known to anyone who regularly follows this blog, is that consciousness and mathematics are just as fundamental to reality as the physical universe, but not in the way that Hoffman argues. I’ve adopted, for better or worse, Roger Penrose’s triumvirate, which he likes to portray in an Escher-like diagram. 

 
I wouldn’t call myself a physicalist when it comes to consciousness, for the simple reason that I don’t believe we can measure it, and despite what Hoffman (and others) often claim, I’m not convinced that it will ever succumb to a mathematical model, in the way that virtually all physical theories do.
 
I left a comment on this video, which was hosted by the ‘Essentia Foundation’, so hopefully, it’s not deleted. Here it is:
 
I agree with him about Godel’s Theorem in its seminal significance to both maths and physics, which is that they are both neverending. However, when he says that ‘reality transcends any mathematical theory’ (3.00) I agree to a point, but I’d argue that mathematics transcends the Universe (known as mathematical Platonism); so in that sense, mathematics transcends reality.
 
The other point, which he never mentions, is that mathematical models of physical phenomena can be wrong – the best example being Ptolemy’s model of the solar system. String theory may well fall into that category – at this stage, we don’t know.
 
When he discusses consciousness being mathematical (4.30): ‘If consciousness is all there is, then mathematical structure is only about consciousness’; which is a premise dressed up as a conclusion, so circular.
 
The problem I’ve always had with Donald Hoffman’s idealism philosophy is that consciousness may exist independently of the Universe; it’s not possible for us to know. But within the Universe itself, evolutionary theory tells us that consciousness came late. Now, I know that he has his own theory of evolution to counter this, but that entails an argument that’s too long to address here.
 
Regarding his argument that spacetime is not fundamental, I know about Nima Arkani-Hamed and his work on amplituhedrons, and to quote: “This is a concrete example of a way in which the physics we normally associate with space-time and quantum mechanics arises from something more basic.” But the something more basic is mathematical, not physical. It’s possible that there was something before spacetime at the very birth of the Universe, but that’s speculative. All our cosmological theories are premised on spacetime.
 
I actually don’t think consciousness can be modelled mathematically, but its neurological underpinnings can, simply because they can be measured. Consciousness itself can’t be measured, only its neurological correlates. In other words, it can’t be measured outside of a brain, which is an object dependent on the Universe’s existence and not the other way round.

The role of dissonance in art, not to mention science and mathematics

 I was given a book for a birthday present just after the turn of the century, titled A Terrible Beauty; The People and Ideas that Shaped the Modern Mind, by Peter Watson. A couple of things worth noting: it covers the history of the 20th Century, but not geo-politically as you might expect. Instead, he writes about the scientific discoveries alongside the arts and cultural innovations, and he talks about both with equal erudition, which is unusual.
 
The reason I mention this, is because I remember Watson talking about the human tendency to push something to its limits and then beyond. He gave examples in science, mathematics, art and music. A good example in mathematics is the adoption of √-1 (giving us ‘imaginary numbers’), which we are taught is impossible, then suddenly it isn’t. The thing is that it allows us to solve problems that were previously impossible in the same way that negative numbers give solutions to arithmetical subtractions that were previously unanswerable. There were no negative numbers in ancient Greece because their mathematics was driven by geometry, and the idea of a negative volume or area made no sense.
 
But in both cases: negative numbers and imaginary numbers; there is a cognitive dissonance that we have to overcome before we can gain familiarity and confidence in using them, or even understanding what they mean in the ‘real world’, which is the problem the ancient Greeks had. Most people reading this have no problem, conceptually, dealing with negative numbers, because, for a start, they’re an integral aspect of financial transactions – I suspect everyone reading this above a certain age has had experience with debt and loans.
 
On the other hand, I suspect a number of readers struggle with a conceptual appreciation of imaginary numbers. Some mathematicians will tell you that the term is a misnomer, and its origin would tend to back that up. Apparently, Rene Descartes coined the term, disparagingly, because, like the ancient Greek’s problem with negative numbers, he believed they had no relevance to the ‘real world’. And Descartes would have appreciated their usefulness in solving problems previously unsolvable, so I expect it would have been a real cognitive dissonance for him.
 
I’ve written an entire post on imaginary numbers, so I don’t want to go too far down that rabbit hole, but I think it’s a good example of what I’m trying to explicate. Imaginary numbers gave us something called complex algebra and opened up an entire new world of mathematics that is particularly useful in electrical engineering. But anyone who has studied physics in the last century is aware that, without imaginary numbers, an entire field of physics, quantum mechanics, would remain indescribable, let alone be comprehensible. The thing is that, even though most people have little or no understanding of QM, every electronic device you use depends on it. So, in their own way, imaginary numbers are just as important and essential to our lives as negative numbers are.
 
You might wonder how I deal with the cognitive dissonance that imaginary numbers induce. In QM, we have, at its most rudimentary level, something called Schrodinger’s equation, which he proposed in 1926 (“It’s not derived from anything we know,” to quote Richard Feynman) and Schrodinger quickly realised it relied on imaginary numbers – he couldn’t formulate it without them. But here’s the thing: Max Born, a contemporary of Schrodinger, formulated something we now call the Born rule that mathematically gets rid of the imaginary numbers (for the sake of brevity and clarity, I’ll omit the details) and this gives the probability of finding the object (usually an electron) in the real world. In fact, without the Born rule, Schrodinger’s equation is next-to-useless, and would have been consigned to the dustbin of history.
 
And that’s relevant, because prior to observing the particle, it’s in a superposition of states, described by Schrodinger’s equation as a wave function (Ψ), which some claim is a mathematical fiction. In other words, you need to get rid (clumsy phrasing, but accurate) of the imaginary component to make it relevant to the reality we actually see and detect. And the other thing is that once we have done that, the Schrodinger equation no longer applies – there is effectively a dichotomy between QM and classical physics (reality), which is called the 'measurement problem’. Roger Penrose gives a good account in this video interview. So, even in QM, imaginary numbers are associated with what we cannot observe.
 
That was a much longer detour than I intended, but I think it demonstrates the dissonance that seems necessary in science and mathematics, and arguably necessary for its progress; plus it’s a good example of the synergy between them that has been apparent since Newton.
 
My original intention was to talk about dissonance in music, and the trigger for this post was a YouTube video by musicologist, Rick Beato (pronounced be-arto), dissecting the Beatles song, Ticket to Ride, which he called, ‘A strange but perfect song’. In fact, he says, “It’s very strange in many ways: it’s rhythmically strange; it’s melodically strange too”. I’ll return to those specific points later. To call Beato a music nerd is an understatement, and he gives a technical breakdown that quite frankly, I can’t follow. I should point out that I’ve always had a good ‘ear’ that I inherited, and I used to sing, even though I can’t read music (neither could the Beatles). I realised quite young that I can hear things in music that others miss. Not totally relevant, but it might explain some things that I will expound upon later.
 
It's a lengthy, in-depth analysis, but if you go to 4.20-5.20, Beato actually introduces the term ‘dissonance’ after he describes how it applies. In effect, there is a dissonance between the notes that John Lennon sings and the chords he plays (on a 12-string guitar). And the thing is that we, the listener, don’t notice – someone (like Beato) has to point it out. Another quote from 15.00: “One of the reasons the Beatles songs are so memorable, is that they use really unusual dissonant notes at key points in the melody.”
 
The one thing that strikes you when you first hear Ticket to Ride is the unusual drum part. Ringo was very inventive and innovative, and became more adventurous, along with his bandmates, on later recordings. The Ticket to Ride drum part has become iconic: everyone knows it and recognises it. There is a good video where Ringo talks about it, along with another equally famous drum part he created. Beato barely mentions it, though right at the beginning, he specifically refers to the song as being ‘rhythmically strange’.
 
A couple of decades ago, can’t remember exactly when, I went and saw an entire Beatles concert put on by a rock band, augmented by orchestral strings and horn parts. It was in 2 parts with an intermission, and basically the 1st half was pre-Sergeant Pepper and the 2nd half, post. I can still remember that they opened the concert with Magical Mystery Tour and it blew me away. The thing is that they went to a lot of trouble to be faithful to the original recordings, and I realised that it was the first time I’d heard their music live, albeit with a cover band. And what immediately struck me was the unusual harmonics and rhythms they employed. Watching Beato’s detailed technical analysis puts this into context for me.
 
Going from imaginary numbers and quantum mechanics to one of The Beatles most popular songs may seem like a giant leap, but it highlights how dissonance is a universal principle for humans, and intrinsic to progression in both art and science.
 
Going back to Watson’s book that I reference in the introduction, another obvious example that he specifically talks about is Picasso’s cubism.
 
In storytelling, it may not be so obvious, and I think modern fiction has been influenced more by cinema than anything else, where the story needs to be more immediate and it needs to flow with minimal description. There is now an expectation that it puts you in the story – what we call immersion.
 
On another level, I’ve noticed a tendency on my part to create cognitive dissonance in my characters and therefore the reader. More than once, I have combined sexual desire with fear, which some may call perverse. I didn’t do this deliberately – a lot of my fiction contains elements I didn’t foresee. Maybe it says something about my own psyche, but I honestly don’t know.

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.
 

What’s inside a black hole?

 The correct answer is no one knows, but I’m going to make a wild, speculative, not fully-informed guess and suggest, possibly nothing. But first, a detour, to provide some context.
 
I came across an interview with very successful, multi-award-winning, Australian-Canadian actor, Pamela Rabe, who is best known (in Australia, at least) for her role in Wentworth (about a fictional female prison). She was interviewed by Benjamin Law in The Age Good Weekend magazine, a few weekends ago, where among many other questions, he asked, Is there a skill you wish you could acquire? She said there were so many, including singing better, speaking more languages and that she wished she was more patient. Many decades ago, I remember someone asking me a similar question, and I can still remember the answer: I said that I wish I was more intelligent, and I think that’s still true.
 
Some people might be surprised by this, and perhaps it’s a good thing I’m not, because I think I would be insufferable. Firstly, I’ve always found myself in the company of people who are much cleverer than me, right from when I started school, and right through my working life. The reason I wish I was more intelligent is that I’ve always been conscious of trying to understand things that are beyond my intellectual abilities. My aspirations don’t match my capabilities.
 
And this brings me to a discussion on black holes, which must, in some respects, represent the limits of what we know about the Universe and maybe what is even possible to know. Not surprisingly, Marcus du Sautoy spent quite a few pages discussing black holes in his excellent book, What We Cannot Know. But there is a short YouTube video by one of the world’s leading exponents on black holes, Kip Thorne, which provides a potted history. I also, not that long ago, read his excellent book, Black Holes and Time Warps; Einstein’s Outrageous Legacy (1994), which gives a very comprehensive history, in which he was not just an observer, but one of the actors.
 
It's worth watching the video because it highlights the role mathematics has played in physics, not only since Galileo, Kepler and Newton, but increasingly so in the 20th Century, following the twin revolutions of quantum mechanics and relativity theory. In fact, relativity theory predicted black holes, yet most scientists (including Einstein, initially) preferred to believe that they couldn’t exist; that Nature wouldn’t allow it.
 
We all suffer from these prejudices, including myself (and even Einstein). I discussed in a recent post how we create mathematical models in an attempt to explain things we observe. But more and more, in physics, we use mathematical models to explain things that we don’t observe, and black holes are the perfect example. If you watch the video interview with Thorne, this becomes obvious, because scientists were gradually won over by the mathematical arguments, before there was any incontrovertible physical evidence that they existed.
 
And since no one can observe what’s inside a black hole, we totally rely on mathematical models to give us a clue. Which brings me to the title of the post. The best known equation in reference to black holes in the Bekenstein-Hawking equation which give us the entropy of a black hole and predicts Hawking radiation. This is yet to be observed, but this is not surprising, as it’s virtually impossible. It’s simply not ‘hot’ enough to distinguish from the CMBR (cosmic microwave background radiation) which permeates the entire universe. 

Here is the formula:

S(BH) = kA/4(lp)^2 

Where S is the entropy of the black hole, A is the surface area of the sphere at the event horizon, and lp is the Planck length given by this formula:

√(Gh/2πc^3) 

Where G is the gravitational constant, h is Planck’s constant and c is the constant for lightspeed.

Hawking liked the idea that it’s the only equation in physics to incorporate the 4 fundamental natural constants: k, G, h and c; in one formula.

So, once again, mathematics predicts something that’s never been observed, yet most scientists believe it to be true. This led to what was called the ‘information paradox’ that all information falling into a black hole would be lost, but what intrigues me is that if a black hole can, in principle, completely evaporate by converting all its mass into radiation, then it infers that the mass is not in fact lost – it must be still there, even if we can’t see it. This means, by inference, that it can’t have disappeared down a wormhole, which is one of the scenarios conjectured.

One of the mathematical models proposed is the 'holographic principle' for black holes, for which I’ll quote directly from Wikipedia, because it specifically references what I’ve already discussed.

The holographic principle was inspired by the Bekenstein bound of black hole thermodynamics, which conjectures that the maximum entropy in any region scales with the radius squared, rather than cubed as might be expected. In the case of a black hole, the insight was that the information content of all the objects that have fallen into the hole might be entirely contained in surface fluctuations of the event horizon. The holographic principle resolves the black hole information paradox within the framework of string theory.

I know this is a long hop to make but what if the horizon not only contains the information but actually contains all the mass. In other words, what if everything is frozen at the event horizon because that’s where time ‘stops’. Most probably not true, and I don’t know enough to make a cogent argument. However, it would mean that the singularity predicted to exist at the centre of a black hole would not include its mass, but only spacetime.

Back in the 70s, I remember reading an article in Scientific American by a philosopher, who effectively argued that a black hole couldn’t exist. Now this was when their purported existence was mostly mathematical, and no one could unequivocally state that they existed physically. I admit I’m hazy about the details but, from what I can remember, he argued that it was self-referencing because it ‘swallowed itself’. Obviously, his argument was much more elaborate than that one-liner suggests. But I do remember thinking his argument flawed and I even wrote a letter to Scientific American challenging it. Basically, I think it’s a case of conflating the language used to describe a phenomenon with the physicality of it.

I only raise it now, because, as a philosopher, I’m just as ignorant of the subject as he was, so I could be completely wrong.

Addendum 1: I was of 2 minds whether to write this, but it kept bugging me - wouldn't leave me alone, so I wrote it down. I've no idea how true it might be, hence all the caveats and qualifications. It's absolutely at the limit of what we can know at this point in time. As I've said before, philosophy exists at the boundary of science and ignorance. It ultimately appealed to my aesthetics and belief in Nature’s aversion to perversity.

Addendum 2: Another reason why I'm most likely wrong is that there is a little known quirk of Newton's theory of gravity that the gravitational 'force' anywhere inside a perfectly symmetrical hollow sphere is zero. So the inside of a black hole exerting zero gravitational force would have to be the ultimate irony, which makes it highly improbable. I've no idea how that relates to the 'holographic principle' for a black hole. But I still don't think all the mass gets sucked into a singularity or down a wormhole. My conjecture is based purely on the idea that 'time' might well become 'zero' at the event horizon, though, from what I've read, no physicist thinks so. From an outsider's perspective, time dilation becomes asymptotically infinite (effective going to zero, but perhaps taking the Universe's lifetime to reach it). In this link, it begs a series of questions that seem to have no definitive answers. The alternative idea is that it's spacetime that 'falls' into a black hole, therefore taking all the mass with it.

Addendum 3: I came across this video by Tibbees (from a year ago), whom I recommend. She cites a book by Carlo Rovelli, White Holes, which is also the title of her video. Now, you can't talk about white holes without talking about black holes; they are just black holes time reversed (as she explicates). We have no evidence they actually exist, unless the Big Bang is a white hole (also mentioned). I have a lot of time for Carlo Rovelli, even though we have philosophical differences (what a surprise). Basically, he argues that, at a fundamental level, time doesn't exist, but it's introduced into the universe as a consequence of entropy (not the current topic). 

Tibbees gives a totally different perspective to my post, which is why I bring it up. Nevertheless, towards the end, she mentions that our view of a hypothetical person (she suggests Rovelli) entering a black hole is that their existence becomes assymptotically infinite. But what, if in this case, what we perceive is what actually happens. Then my scenario makes sense. No one else believes that, so it's probably incorrect.

Addendum 4: Victor T Toth, whom even Mark John Fernee defers to (on Quora), when it comes to cosmology and gravity, has said more than once, that 'the event horizon is always in your future', which infers you never reach it. This seems to contradict the prevailing view among physicists that, while that's true for another 'observer' observing 'you' (assuming you're the one falling into a black hole), from 'your' perspective you could cross the event horizon without knowing you have (see the contradiction). This is the conventional, prevailing view among physicists. To my knowledge, Toth has never addressed this apparent contradiction specifically.

However, if one follows Toth's statement to its logical conclusion, 'you' would approach the event horizon asymptotically, which is what I'm speculating. In which case, everything that falls into a black hole would accumulate at the event horizon. The thing is that gravity determines the 'true time' (τ) for a free falling object, and if τ became zero at the event horizon, then everything I've said makes sense. The thing is I really don't know enough physics to back up my conjecture with mathematics.

Addendum 5: Possibly the most important addendum to this post, in that it provides yet another plausible scenario based on what we currently know, and is rather eruditely expounded upon by someone on Quora calling himself 'The Physics Detective' (John Duffield). Of course, I've heard of the 'Firewall' explanation, without knowing if it's true or not. But I suspect no one does.

Image by W H Freeman and company, publishers of Gravitation

Addendum 6: I don't think I've ever written so many addendums to a post, which demonstrates how equivocal and unconvinced I am by my own arguments. It's symptomatic of our ignorance, and mine in particular, on this subject.

So I'm going to renege and go back to an earlier post I wrote, where I align myself with Kip Thorne, who is an actual expert on this matter.

Do we make reality?

 I’ve read 2 articles, one in New Scientist (12 Oct 2024) and one in Philosophy Now (Issue 164, Oct/Nov 2024), which, on the surface, seem unrelated, yet both deal with human exceptionalism (my term) in the context of evolution and the cosmos at large.
 
Staring with New Scientist, there is an interview with theoretical physicist, Daniele Oriti, under the heading, “We have to embrace the fact that we make reality” (quotation marks in the original). In some respects, this continues on with themes I raised in my last post, but with different emphases.
 
This helps to explain the title of the post, but, even if it’s true, there are degrees of possibilities – it’s not all or nothing. Having said that, Donald Hoffman would argue that it is all or nothing, because, according to him, even ‘space and time don’t exist unperceived’. On the other hand, Oriti’s argument is closer to Paul Davies’ ‘participatory universe’ that I referenced in my last post.
 
Where Oriti and I possibly depart, philosophically speaking, is that he calls the idea of an independent reality to us ‘observers’, “naïve realism”. He acknowledges that this is ‘provocative’, but like many provocative ideas it provides food-for-thought. Firstly, I will delineate how his position differs from Hoffman’s, even though he never mentions Hoffman, but I think it’s important.
 
Both Oriti and Hoffman argue that there seems to be something even more fundamental than space and time, and there is even a recent YouTube video where Hoffman claims that he’s shown mathematically that consciousness produces the mathematical components that give rise to spacetime; he has published a paper on this (which I haven’t read). But, in both cases (by Hoffman and Oriti), the something ‘more fundamental’ is mathematical, and one needs to be careful about reifying mathematical expressions, which I once discussed with physicist, Mark John Fernee (Qld University).
 
The main issue I have with Hoffman’s approach is that space-time is dependent on conscious agents creating it, whereas, from my perspective and that of most scientists (although I’m not a scientist), space and time exists external to the mind. There is an exception, of course, and that is when we dream.
 
If I was to meet Hoffman, I would ask him if he’s heard of proprioception, which I’m sure he has. I describe it as the 6th sense we are mostly unaware of, but which we couldn’t live without. Actually, we could, but with great difficulty. Proprioception is the sense that tells us where our body extremities are in space, independently of sight and touch. Why would we need it, if space is created by us? On the other hand, Hoffman talks about a ‘H sapiens interface’, which he likens to ‘desktop icons on a computer screen’. So, somehow our proprioception relates to a ‘spacetime interface’ (his term) that doesn’t exist outside the mind.
 
A detour, but relevant, because space is something we inhabit, along with the rest of the Universe, and so is time. In relativity theory there is absolute space-time, as opposed to absolute space and time separately. It’s called the fabric of the universe, which is more than a metaphor. As Viktor Toth points out, even QFT seems to work ‘just fine’ with spacetime as its background.
 
We can do quantum field theory just fine on the curved spacetime background of general relativity.
 
[However] what we have so far been unable to do in a convincing manner is turn gravity itself into a quantum field theory.
 
And this is where Oriti argues we need to find something deeper. To quote:
 
Modern approaches to quantum gravity say that space-time emerges from something deeper – and this could offer a new foundation for physical laws.
 
He elaborates: I work with quantum gravity models in which you don’t start with a space-time geometry, but from more abstract “atomic” objects described in purely mathematical language. (Quotation marks in the original.)
 
And this is the nub of the argument: all our theories are mathematical models and none of them are complete, in as much as they all have limitations. If one looks at the history of physics, we have uncovered new ‘laws’ and new ‘models’ when we’ve looked beyond the limitations of an existing theory. And some mathematical models even turned out to be incorrect, despite giving answers to what was ‘known’ at the time. The best example being Ptolemy’s Earth-centric model of the solar system. Whether string theory falls into the same category, only future historians will know.
 
In addition, different models work at different scales. As someone pointed out (Mile Gu at the University of Queensland), mathematical models of phenomena at one scale are different to mathematical models at an underlying scale. He gave the example of magnetism, demonstrating that mathematical modelling of the magnetic forces in iron could not predict the pattern of atoms in a 3D lattice as one might expect. In other words, there should be a causal link between individual atoms and the overall effect, but it could not be determined mathematically. To quote Gu: “We were able to find a number of properties that were simply decoupled from the fundamental interactions.” Furthermore, “This result shows that some of the models scientists use to simulate physical systems have properties that cannot be linked to the behaviour of their parts.”
 
This makes me sceptical that we will find an overriding mathematical model that will entail the Universe at all scales, which is what theories of quantum gravity attempt to do. One of the issues that some people raise is that a feature of QM is superposition, and the superposition of a gravitational field seems inherently problematic.
 
Personally, I think superposition only makes sense if it’s describing something that is yet to happen, which is why I agree with Freeman Dyson that QM can only describe the future, which is why it only gives us probabilities.
 
Also, in quantum cosmology, time disappears (according to Paul Davies, among others) and this makes sense (to me), if it’s attempting to describe the entire universe into the future. John Barrow once made a similar point, albeit more eruditely.
 
Getting off track, but one of the points that Oriti makes is whether the laws and the mathematics that describes them are epistemic or ontic. In other words, are they reality or just descriptions of reality. I think it gets blurred, because while they are epistemic by design, there is still an ontology that exists without them, whereas Oriti calls that ‘naïve realism’. He contends that reality doesn’t exist independently of us. This is where I always cite Kant: that we may never know the ‘thing-in-itself,’ but only our perception of it. Where I diverge from Kant is that the mathematical models are part of our perception. Where I depart from Oriti is that I argue there is a reality independently of us.
 
Both QM and relativity theory are observer-dependent, which means they could both be describing an underlying reality that continually eludes us. Whereas Oriti argues that ‘reality is made by our models, not just described by them’, which would make it subjective.
 
As I pointed out in my last post, there is an epistemological loop, whereby the Universe created the means to understand itself, through us. Whether there is also an ontological loop as both Davies and Oriti infer, is another matter: do we determine reality through our quantum mechanical observations? I will park that while I elaborate on the epistemic loop.
 
And this finally brings me to the article in Philosophy Now by James Miles titled, We’re as Smart as the Universe gets. He argues that, from an evolutionary perspective, there is a one-in-one-billion possibility that a species with our cognitive abilities could arise by natural selection, and there is no logical reason why we would evolve further, from an evolutionary standpoint. I have touched on this before, where I pointed out that our cultural evolution has overtaken our biological evolution and that would also happen to any other potential species in the Universe who developed cognitive abilities to the same level. Dawkins coined the term, ‘meme’, to describe cultural traits that have ‘survived’, which now, of course, has currency on social media way beyond its original intention. Basically, Dawkins saw memes as analogous to genes, which get selected; not by a natural process but by a cultural process.
 
I’ve argued elsewhere that mathematical theorems and scientific theories are not inherently memetic. This is because they are chosen because they are successful, whereas memes are successful because they are chosen. Nevertheless, such theorems and theories only exist because a culture has developed over millennia which explores them and builds on them.
 
Miles talks about ‘the high intelligence paradox’, which he associates with Darwin’s ‘highest and most interesting problem’. He then discusses the inherent selection advantage of co-operation, not to mention specialisation. He talks about the role that language has played, which is arguably what really separates us from other species. I’ve argued that it’s our inherent ability to nest concepts within concepts ad-infinitum (which is most obvious in our facility for language, like I’m doing now) that allows us to, not only tell stories, compose symphonies, explore an abstract mathematical landscape, but build motor cars, aeroplanes and fly men to the moon. Are we the only species in the Universe with this super-power? I don’t know, but it’s possible.
 
There are 2 quotes I keep returning to:
 
The most incomprehensible thing about the Universe is that it’s comprehensible. (Einstein)
 
The Universe gave rise to consciousness and consciousness gives meaning to the Universe. (Wheeler)
 
I haven’t elaborated, but Miles makes the point, while referencing historical antecedents, that there appears no evolutionary 'reason’ that a species should make this ‘one-in-one-billion transition’ (his nomenclature). Yet, without this transition, the Universe would have no meaning that could be comprehended. As I say, that’s the epistemic loop.
 
As for an ontic loop, that is harder to argue. Photons exist in zero time, which is why I contend they are always in the future of whatever they interact with, even if they were generated in the CMBR some 13.5 billion years ago. So how do we resolve that paradox? I don’t know, but maybe that’s the link that Davies and Oriti are talking about, though neither of them mention it. But here’s the thing: when you do detect such a photon (for which time is zero) you instantaneously ‘see’ back to 380,000 years after the Universe’s birth.

Science and religion meet at the boundary of humanity’s ignorance

 I watched a YouTube debate (90 mins) between Sir Roger Penrose and William Lane Craig, and, if I’m honest, I found it a bit frustrating because I wish I was debating Craig instead of Penrose. I also think it would have been more interesting if Craig debated someone like Paul Davies, who is more philosophically inclined than Penrose, even though Penrose is more successful as a scientist, and as a physicist, in particular.
 
But it was set up as an atheist versus theist debate between 2 well known personalities, who were mutually respectful and where there was no animosity evident at all. I confess to having my own biases, which would be obvious to any regular reader of this blog. I admit to finding Craig arrogant and a bit smug in his demeanour, but to be fair, he was on his best behaviour, and perhaps he’s matured (or perhaps I have) or perhaps he adapts to whoever he’s facing. When I call it a debate, it wasn’t very formal and there wasn’t even a nominated topic. I felt the facilitator or mediator had his own biases, but I admit it would be hard to find someone who didn’t.
 
Penrose started with his 3 worlds philosophy of the physical, the mental and the abstract, which has long appealed to me, though most scientists and many philosophers would contend that the categorisation is unnecessary, and that everything is physical at base. Penrose proposed that they present 3 mysteries, though the mysteries are inherent in the connections between them rather than the categories themselves. This became the starting point of the discussion.
 
Craig argued that the overriding component must surely be ‘mind’, whereas Penrose argued that it should be the abstract world, specifically mathematics, which is the position of mathematical Platonists (including myself). Craig pointed out that mathematics can’t ‘create’ the physical, (which is true) but a mind could. As the mediator pointed out (as if it wasn’t obvious) said mind could be God. And this more or less set the course for the remainder of the discussion, with a detour to Penrose’s CCC theory (Conformal Cyclic Cosmology).
 
I actually thought that this was Craig’s best argument, and I’ve written about it myself, in answer to a question on Quora: Did math create the Universe? The answer is no, nevertheless I contend that mathematics is a prerequisite for the Universe to exist, as the laws that allowed the Universe to evolve, in all its facets, are mathematical in nature. Note that this doesn’t rule out a God.
 
Where I would challenge Craig, and where I’d deviate from Penrose, is that we have no cognisance of who this God is or even what ‘It’ could be. Could not this God be the laws of the Universe themselves? Penrose struggled with this aspect of the argument, because, from a scientific perspective, it doesn’t tell us anything that we can either confirm or falsify. I know from previous debates that Craig has had, that he would see this as a win. A scientist can’t refute his God’s existence, nor can they propose an alternative, therefore it’s his point by default.
 
This eventually led to a discussion on the ‘fine-tuning’ of the Universe, which in the case of entropy, is what led Penrose to formulate his CCC model of the Universe. Of course, the standard alternative is the multiverse and the anthropic principle, which, as Penrose points out, is also applicable to his CCC model, where you have an infinite sequence of universes as opposed to an infinity of simultaneous ones, which is the orthodox response among cosmologists.
 
This is where I would have liked to have seen Paul Davies respond, because he’s an advocate of John Wheeler’s so-called ‘participatory Universe’, which is effectively the ‘strong anthropic principle’ as opposed to the ‘weak anthropic principle’. The weak anthropic principle basically says that ‘observers’ (meaning us) can only exist in a universe that allows observers to exist – a tautology. Whereas the strong anthropic principle effectively contends that the emergence of observers is a necessary condition for the Universe to exist (the observers don’t have to be human). Basically, Wheeler was an advocate of a cosmic, acausal (backward-in-time) link from conscious observers to the birth of the Universe. I admit this appeals to me, but as Craig would expound, it’s a purely metaphysical argument, and so is the argument for God.
 
The other possibility that is very rarely expressed, is that God is the end result of the Universe rather than its progenitor. In other words, the ‘mind’ that Craig expounded upon is a consequence of all of us. This aligns more closely with the Hindu concept of Brahman or a Buddhist concept of collective karma – we get the God we deserve. Erwin Schrodinger, who studied the Upanishads, discusses Brahman as a pluralistic ‘mind’ in What is Life?. (Note that in Hinduism, the soul or Atman is a part of Brahman). My point would be that the Judea-Christian-Islamic God does not have a monopoly on Craig’s overriding ‘mind’ concept.
 
A recurring theme on this blog is that there will always be mysteries – we can never know everything – and it’s an unspoken certitude that there will forever be knowledge beyond our cognition. The problem that scientists sometimes have, but are reluctant to admit, is that we can’t explain everything, even though we keep explaining more by the generation. And the problem that theologians sometimes have is that our inherent ignorance is neither ‘proof’ nor ‘evidence’ that there is a ‘creator’ God.
 
I’ve argued elsewhere that a belief in God is purely a subjective and emotional concept, which one then rationalises with either cultural references or as an ultimate explanation for our existence.

Addendum: I like this quote, albeit out of context, from Spinoza:: "The sum of the natural and physical laws of the universe and certainly not an individual entity or creator".

How scale demonstrates that mathematics is intrinsically entailed in the Universe

 I momentarily contemplated another title: Is the Planck limit an epistemology or an ontology? Because that’s basically the topic of a YouTube video that’s the trigger for this post. I wrote a post some time ago where I discussed whether the Universe is continuous or discrete, and basically concluded that it was continuous. Based on what I’ve learned from this video, I might well change my position. But I should point out that my former opposition was based more on the idea that it could be quantised into ‘bits’ of information, whereas now I’m willing to acknowledge that it could be granular at the Planck scale, which I’ll elaborate on towards the end. I still don’t think that the underlying reality of the Universe is in ‘bits’ of information, therefore potentially created and operated by a computer.
 
Earlier this year, I discussed the problem of reification of mathematics so I want to avoid that if possible. By reification, I mean making a mathematical entity reality. Basically, physics works by formulating mathematical models that we then compare to reality through observations. But as Freeman Dyson pointed out, the wave function (Ψ), for example, is a mathematical entity and not a physical entity, which is sometimes debated. The fact is that if it does exist physically, it’s never observed, and my contention is that it ‘exists’ in the future; a view that is consistent with Dyson’s own philosophical viewpoint that QM can only describe the future and not the past.
 
And this brings me to the video, which has nothing to say about wave functions or reified mathematical entities, but uses high school mathematics to explore such esoteric and exotic topics as black holes and quantum gravity. There is one step involving integral calculus, which is about as esoteric as the maths becomes, and if you allow that 1/∞ = 0, it leads to the formula for the escape velocity from any astronomical body (including Earth). Note that the escape velocity literally allows an object to escape a gravitational field to infinity (∞). And the escape velocity for a black hole is c (the speed of light).
 
All the other mathematics is basic algebra using some basic physics equations, like Newton’s equation for gravity, Planck’s equation for energy, Heisenberg’s Uncertainty Principle using Planck’s Constant (h), Einstein’s famous equation for the equivalence of energy and mass, and the equation for the Coulomb Force between 2 point electric charges (electrons). There is also the equation for the Schwarzschild radius of a black hole, which is far easier to derive than you might imagine (despite the fact that Schwarzschild originally derived it from Einstein’s field equations).
 
Back in May 2019, I wrote a post on the Universe’s natural units, which involves the fundamental natural constants, h, c and G. This was originally done by Planck himself, which I describe in that post, while providing a link to a more detailed exposition. In the video (embedded below), the narrator takes a completely different path to deriving the same Planck units before describing a method that Planck himself would have used. In so doing, he explains how at the Planck level, space and time are not only impossible to observe, even in principle, but may well be impossible to remain continuous in reality. You need to watch the video, as he explains it far better than I can, just using high school mathematics.
 
Regarding the title I chose for this post, Roger Penrose’s Conformal Cyclic Cosmology (CCC) model of the Universe, exploits the fact that a universe without matter (just radiation) is scale invariant, which is essential for the ‘conformal’ part of his theory. However, that all changes when one includes matter. I’ve argued in other posts that different forces become dominant at different scales, from the cosmological to the subatomic. The point made in this video is that at the Planck scale all the forces, including gravity, become comparable. Now, as I pointed out at the beginning, physics is about applying mathematical models and comparing them to reality. We can’t, and quite possibly never will, be able to observe reality at the Planck scale, yet the mathematics tells us that it’s where all the physics we currently know is compatible. It tells me that not only is the physics of the Universe scale-dependent, but it's also mathematically dependent (because scale is inherently mathematical). In essence, the Universe’s dynamics are determined by mathematical parameters at all scales, including the Planck scale.
 
Note that the mathematical relationships in the video use ~ not = which means that they are approximate, not exact. But this doesn’t detract from the significance that 2 different approaches arrive at the same conclusion, which is that the Planck scale coincides with the origin of the Universe incorporating all forces equivalently.
 
 
Addendum: I should point out that Viktor T Toth, who knows a great deal more about this than me, argues that there is, in fact, no limit to what we can measure in principle. Even the narrator in the video frames his conclusion cautiously and with caveats. In other words, we are in the realm of speculative physics. Nevertheless, I find it interesting to contemplate where the maths leads us.

More on radical ideas

 As you can tell from the title, this post carries on from the last one, because I got a bit bogged down on one issue, when I really wanted to discuss more. One of the things that prompted me was watching a 1hr presentation by cosmologist, Claudia de Rahm, whom I’ve mentioned before, when I had the pleasure of listening to an on-line lecture she gave, care of New Scientist, during the COVID lockdown.
 
Claudia’s particular field of study is gravity, and, by her own admission, she has a ‘crazy idea’. Now here’s the thing: I meet a lot of people on Quora and in the blogosphere, who like me, live (in a virtual sense) on the fringes of knowledge rather than as academic or professional participants. And what I find is that they often have an almost zealous confidence in their ideas. To give one example, I recently came across someone who argued quite adamantly that the Universe is static, not expanding, and has even written a book on the subject. This is contrary to virtually everyone else I’m aware of who works in the field of cosmology and astrophysics. And I can’t help but compare this to Claudia de Rahm who is well aware that her idea is ‘crazy’, even though she’s fully qualified to argue it.
 
In other words, it’s a case of the more you know about a subject, the less you claim to know, because experts are more aware of their limitations than non-experts. I should point out, in case you didn’t already know, I’m not one of the experts.
 
Specifically, Claudia’s crazy idea is that not only are there gravitational waves, but gravitons and that gravitons have an extremely tiny amount of mass, which would alter the effect of gravity at very long range. I should say that at present, the evidence is against her, because if she’s right, gravity waves would travel not at the speed of light, as predicted by Einstein, but ever-so-slightly less than light.
 
Freeman Dyson, by the way, has argued that if gravitons do exist, they would be impossible to detect, but if Claudia is right, then they would be.
 
In her talk, Claudia also discusses the vacuum energy, which according to particle physics, should be 28 orders of magnitude greater than the relativistic effect of ‘dark energy’. She calls it ‘the biggest discrepancy in the entire history of science’. This suggests that there is something rotten in the state of theoretical physics, along with the fact, that what we can physically observe, only accounts for 5% of the Universe.
 
It should be pointed out that at the end of the 19th Century no one saw or predicted the 2 revolutions in physics that were just around the corner – relativity theory and quantum mechanics. They were an example of what Thomas Kuhn called The Structure of Scientific Revolutions (the title of his book expounding on this). And I’d suggest that these current empirical aberrations in cosmology are harbingers of the next Kuhnian revolution.
 
Roger Penrose, whom I’ve referenced a number of times on this blog, is someone else with some ‘crazy’ ideas compared to the status quo, for which I admire him even if I don’t agree with him. One of Penrose’s hobby horses is his own particular inference from Godel’s Incompleteness Theorem, which he learned as a graduate (under Steen, at Cambridge) and which he discusses in this video. He argues that it provides evidence that humans don’t think like computers. If one takes the example of Riemann’s Hypothesis (really a conjecture) we know that a computer can’t tell us if it’s true or not (my example, not Penrose’s).* However, most mathematicians believe it is true, and it would be an enormous shock if it was proven untrue, or a contra-example was found by a computer. This is the case with other conjectures that have been proven true, like Fermat’s Last Theorem and Poincare’s conjecture. Penrose’s point, if I understand him correctly, is that it takes a human mind and not a computer to make this leap into the unknown and grasp a ‘truth’ out of the aether.
 
Anyone who has engaged in some artistic endeavour can identify with this, even if it’s not mathematical truths they are seeking but the key to unravelling a plot in a story.
 
Penrose makes the point in the video that he’s a ‘visual’ person, which he thinks is unusual in his field. Penrose is an excellent artist, by the way, and does all his own graphics. This is something else I can identify with, as I was quite precocious as a very young child at drawing (I could draw in perspective, though no one taught me) even though it never went anywhere.
 
Finally, some crazy ideas of my own. I’ve pointed out on other posts that I have a predilection (for want of a better term) for Kant’s philosophical proposition that we can never know the ‘thing-in-itself’ but only a perception of it.
 
With this in mind, I contend that this philosophical premise not only applies to what we can physically detect via instruments, but what we theoretically infer from the mathematics we use to explore nature. As heretical an idea as it may seem, I argue that mathematics is yet another 'instrument' we use to probe the secrets of the Universe. Quantum mechanics and relativity theory being the most obvious.
 
As I’ve tried to expound on other posts, relativity theory is observer-dependent, in as much as different observers will both measure and calculate different values of time and space, dependent on their specific frame of reference. I believe this is a pertinent example of Kant’s proposition that the thing-in-itself escapes our perception. In particular, physicists (including Penrose) will tell you that events that are ostensibly simultaneous to us (in a galaxy far, far away) will be perceived as both past and future by 2 observers who are simply crossing a street in opposite directions. I’ve written about this elsewhere as ‘the impossible thought experiment’.
 
The fact is that relativity theory rules out the event being observed at all. In other words, simultaneous events can’t be observed (according to relativity). For this reason, virtually all physicists will tell you that simultaneity is an illusion – there is no universal now.
 
But here’s the thing: if there is an edge in either space or time, it can only be observed from outside the Universe. Relativity theory, logically enough, can only tell us what we can observe from within the Universe.
 
But to extend this crazy idea, what’s stopping the Universe existing within a higher dimension that we can’t perceive. Imagine being a fish and you spend your entire existence in a massive body of water, which is your entire universe. But then one day you are plucked out of that environment and you suddenly become aware that there is another, even bigger universe that exists right alongside yours.
 
There is a tendency for us to think that everything that exists we can learn and know about – it’s what separates us from every other living thing on the planet. But perhaps there are other dimensions, or even worlds, that lie forever beyond our comprehension.

*Footnote: Actually, Penrose in his book, The Emperor’s New Mind, discusses this in depth and at length over a number of chapters. He makes the point that Turing’s ‘proof’ that it’s impossible to predict whether a machine attempting to compute all the Riemann zeros (for example) will stop, is a practical demonstration of the difference between ‘truth’ and ‘proof’ (as Godel’s Incompleteness Theorem tell us). Quite simply, if the theorem is true, the computer will never stop, so it can never be proven algorithmically. It can only be proven (or disproven) if one goes ‘outside the [current] rules’ to use Penrose’s own nomenclature.

Radical ideas

 It’s hard to think of anyone I admire in physics and philosophy who doesn’t have at least one radical idea. Even Richard Feynman, who avoided hyperbole and embraced doubt as part of his credo: "I’d rather have doubt and be uncertain, than be certain and wrong."
 
But then you have this quote from his good friend and collaborator, Freeman Dyson:


Thirty-one years ago, Dick Feynman told me about his ‘sum over histories’ version of quantum mechanics. ‘The electron does anything it likes’, he said. ‘It goes in any direction at any speed, forward and backward in time, however it likes, and then you add up the amplitudes and it gives you the wave-function.’ I said, ‘You’re crazy.’ But he wasn’t.
 
In fact, his crazy idea led him to a Nobel Prize. That exception aside, most radical ideas are either still-born or yet to bear fruit, and that includes mine. No, I don’t compare myself to Feynman – I’m not even a physicist - and the truth is I’m unsure if I even have an original idea to begin with, be they radical or otherwise. I just read a lot of books by people much smarter than me, and cobble together a philosophical approach that I hope is consistent, even if sometimes unconventional. My only consolation is that I’m not alone. Most, if not all, the people smarter than me, also hold unconventional ideas.
 
Recently, I re-read Robert M. Pirsig’s iconoclastic book, Zen and the Art of Motorcycle Maintenance, which I originally read in the late 70s or early 80s, so within a decade of its publication (1974). It wasn’t how I remembered it, not that I remembered much at all, except it had a huge impact on a lot of people who would never normally read a book that was mostly about philosophy, albeit disguised as a road-trip. I think it keyed into a zeitgeist at the time, where people were questioning everything. You might say that was more the 60s than the 70s, but it was nearly all written in the late 60s, so yes, the same zeitgeist, for those of us who lived through it.
 
Its relevance to this post is that Pirsig had some radical ideas of his own – at least, radical to me and to virtually anyone with a science background. I’ll give you a flavour with some selective quotes. But first some context: the story’s protagonist, whom we assume is Pirsig himself, telling the story in first-person, is having a discussion with his fellow travellers, a husband and wife, who have their own motorcycle (Pirsig is travelling with his teenage son as pillion), so there are 2 motorcycles and 4 companions for at least part of the journey.
 
Pirsig refers to a time (in Western culture) when ghosts were considered a normal part of life. But then introduces his iconoclastic idea that we have our own ghosts.
 
Modern man has his own ghosts and spirits too, you know.
The laws of physics and logic… the number system… the principle of algebraic substitution. These are ghosts. We just believe in them so thoroughly they seem real.
 
Then he specifically cites the law of gravity, saying provocatively:
 
The law of gravity and gravity itself did not exist before Isaac Newton. No other conclusion makes sense.
And what that means, is that the law of gravity exists nowhere except in people’s heads! It’s a ghost! We are all of us very arrogant and conceited about running down other people’s ghosts but just as ignorant and barbaric and superstitious about our own.
Why does everybody believe in the law of gravity then?
Mass hypnosis. In a very orthodox form known as “education”.
 
He then goes from the specific to the general:
 
Laws of nature are human inventions, like ghosts. Laws of logic, of mathematics are also human inventions, like ghosts. The whole blessed thing is a human invention, including the idea it isn’t a human invention. (His emphasis)
 
And this is philosophy in action: someone challenges one of your deeply held beliefs, which forces you to defend it. Of course, I’ve argued the exact opposite, claiming that ‘in the beginning there was logic’. And it occurred to me right then, that this in itself, is a radical idea, and possibly one that no one else holds. So, one person’s radical idea can be the antithesis of someone else’s radical idea.
 
Then there is this, which I believe holds the key to our disparate points of view:
 
We believe the disembodied 'words' of Sir Isaac Newton were sitting in the middle of nowhere billions of years before he was born and that magically he discovered these words. They were always there, even when they applied to nothing. Gradually the world came into being and then they applied to it. In fact, those words themselves were what formed the world. (again, his emphasis)
 
Note his emphasis on 'words', as if they alone make some phenomenon physically manifest.
 
My response: don’t confuse or conflate the language one uses to describe some physical entity, phenomena or manifestation with what it describes. The natural laws, including gravity, are mathematical in nature, obeying sometimes obtuse and esoteric mathematical relationships, which we have uncovered over eons of time, which doesn’t mean they only came into existence when we discovered them and created the language to describe them. Mathematical notation only exists in the mind, correct, including the number system we adopt, but the mathematical relationships that notation describes, exist independently of mind in the same way that nature’s laws do.
 
John Barrow, cosmologist and Fellow of the Royal Society, made the following point about the mathematical ‘laws’ we formulated to describe the first moments of the Universe’s genesis (Pi in the Sky, 1992).
 
Specifically, he says our mathematical theories describing the first three minutes of the Universe predict specific ratios of the earliest ‘heavier’ elements: deuterium, 2 isotopes of helium and lithium, which are 1/1000, 1/1000, 22 and 1/100,000,000 respectively; with the remaining (roughly 78%) being hydrogen. And this has been confirmed by astronomical observations. He then makes the following salient point:



It confirms that the mathematical notions that we employ here and now apply to the state of the Universe during the first three minutes of its expansion history at which time there existed no mathematicians… This offers strong support for the belief that the mathematical properties that are necessary to arrive at a detailed understanding of events during those first few minutes of the early Universe exist independently of the presence of minds to appreciate them.
 
As you can see this effectively repudiates Pirsig’s argument; but to be fair to Pirsig, Barrow wrote this almost 2 decades after Pirsig’s book.
 
In the same vein, Pirsig then goes on to discuss Poincare’s Foundations of Science (which I haven’t read), specifically talking about Euclid’s famous fifth postulate concerning parallel lines never meeting, and how it created problems because it couldn’t be derived from more basic axioms and yet didn’t, of itself, function as an axiom. Euclid himself was aware of this, and never used it as an axiom to prove any of his theorems.
 
It was only in the 19th Century, with the advent of Riemann and other non-Euclidean geometries on curved surfaces that this was resolved. According to Pirsig, it led Poincare to question the very nature of axioms.
 
Are they synthetic a priori judgements, as Kant said? That is, do they exist as a fixed part of man’s consciousness, independently of experience and uncreated by experience? Poincare thought not…
Should we therefore conclude that the axioms of geometry are experimental verities? Poincare didn’t think that was so either…
Poincare concluded that the axioms of geometry are conventions, our choice among all possible conventions is guided by experimental facts, but it remains free and is limited only by the necessity of avoiding all contradiction.
 
I have my own view on this, but it’s worth seeing where Pirsig goes with it:
 
Then, having identified the nature of geometric axioms, [Poincare] turned to the question, Is Euclidean geometry true or is Riemann geometry true?
He answered, The question has no meaning.
[One might] as well as ask whether the metric system is true and the avoirdupois system is false; whether Cartesian coordinates are true and polar coordinates are false. One geometry can not be more true than another; it can only be more convenient. Geometry is not true, it is advantageous.
 
I think this is a false analogy, because the adoption of a system of measurement (i.e. units) and even the adoption of which base arithmetic one uses (decimal, binary, hexadecimal being the most common) are all conventions.
 
So why wouldn’t I say the same about axioms? Pirsig and Poincare are right in as much that both Euclidean and Riemann geometry are true because they’re dependent on the topology that one is describing. They are both used to describe physical phenomena. In fact, in a twist that Pirsig probably wasn’t aware of, Einstein used Riemann geometry to describe gravity in a way that Newton could never have envisaged, because Newton only had Euclidean geometry at his disposal. Einstein formulated a mathematical expression of gravity that is dependent on the geometry of spacetime, and has been empirically verified to explain phenomena that Newton couldn’t. Of course, there are also limits to what Einstein’s equations can explain, so there are more mathematical laws still to uncover.
 
But where Pirsig states that we adopt the axiom that is convenient, I contend that we adopt the axiom that is necessary, because axioms inherently expand the area of mathematics we are investigating. This is a consequence of Godel’s Incompleteness Theorem that states there are limits to what any axiom-based, consistent, formal system of mathematics can prove to be true. Godel himself pointed out that that the resolution lies in expanding the system by adopting further axioms. The expansion of Euclidean to non-Euclidean geometry is a case in point. The example I like to give is the adoption of √-1 = i, which gave us complex algebra and the means to mathematically describe quantum mechanics. In both cases, the axioms allowed us to solve problems that had hitherto been impossible to solve. So it’s not just a convenience but a necessity.
 
I know I’ve belaboured a point, but both of these: non-Euclidean geometry and complex algebra; were at one time radical ideas in the mathematical world that ultimately led to radical ideas: general relativity and quantum mechanics; in the scientific world. Are they ghosts? Perhaps ghost is an apt metaphor, given that they appear timeless and have outlived their discoverers, not to mention the rest of us. Most physicists and mathematicians tacitly believe that they not only continue to exist beyond us, but existed prior to us, and possibly the Universe itself.
 
I will briefly mention another radical idea, which I borrowed from Schrodinger but drew conclusions that he didn’t formulate. That consciousness exists in a constant present, and hence creates the psychological experience of the flow of time, because everything else becomes the past as soon as it happens. I contend that only consciousness provides a reference point for past, present and future that we all take for granted.

It all started with Euclid

 I’ve mentioned Euclid before, but this rumination was triggered by a post on Quora that someone wrote about Plato, where they argued, along with another contributor, that Plato is possibly overrated because he got a lot of things wrong, which is true. Nevertheless, as I’ve pointed out in other posts, his Academy was effectively the origin of Western philosophy, science and mathematics. It was actually based on the Pythagorean quadrivium of geometry, arithmetic, astronomy and music.
 
But Plato was also a student and devoted follower of Socrates and the mentor of Aristotle, who in turn mentored Alexander the Great. So Plato was a pivotal historical figure and without his writings, we probably wouldn’t know anything about Socrates. In the same way that, without Paul, we probably wouldn’t know anything about Jesus. (I’m sure a lot of people would find that debatable, but, if so, it’s a debate for another post.)
 
Anyway, I mentioned Euclid in my own comment (on Quora), who was the Librarian at Alexandria around 300BC, and thus a product of Plato’s school of thought. Euclid wrote The Elements, which I contend is arguably the most important book written in the history of humankind – more important than any religious text, including the Bible, Homer’s Iliad and the Mahabharata, which, I admit, is quite a claim. It's generally acknowledged as the most copied text in the secular world. In fact, according to Wikipedia:
 
It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first printing in 1482.
 
Euclid was revolutionary in one very significant way: he was able to demonstrate what ‘truth’ was, using pure logic, albeit in a very abstract and narrow field of inquiry, which is mathematics.
 
Before then, and in other cultures, truth was transient and subjective and often prescribed by the gods. But Euclid changed all that, and forever. I find it extraordinary that I was examined on Euclid’s theorems in high school in the 20th Century.
 
And this mathematical insight has become, millennia later, a key ingredient (for want of a better term) in the hunt for truths in the physical world. In the 20th Century, in what has become known as the Golden Age of Physics, the marriage between mathematics and scientific inquiry at all scales, from the cosmic to the infinitesimal, has uncovered deeply held secrets of nature that the Pythagoreans, and Euclid for that matter, could never have dreamed of. Look no further than quantum mechanics (QM) and the General Theory of Relativity (GR). Between these 2 iconic developments, they underpin every theory we currently have in physics, and they both rely on mathematics that was pivotal in the development of the theories from the outset. In other words, without the mathematics of complex algebra and Riemann geometry respectively, these theories would have been stillborn.
 
I like to quote Richard Feynman from his book, The Character of Physical Law, in a chapter titled, The Relation of Mathematics to Physics:
 
…what turns out to be true is that the more we investigate, the more laws we find, and the deeper we penetrate nature, the more this disease persists. Every one of our laws is a purely mathematical statement in rather complex and abstruse mathematics... Why? I have not the slightest idea. It is only my purpose to tell you about this fact.
 
The strange thing about physics is that for the fundamental laws we still need mathematics.
 
Physicists cannot make a conversation in any other language. If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. She offers her information only in one form.
 
And this has only become more evident since Feynman wrote those words.
 
There was another revolution in the 20th Century, involving Alan Turing, Alonso Church and Kurt Godel; this time involving mathematics itself. Basically, each of these independently demonstrated that some mathematical truths were elusive to proof. Some mathematical conjectures could not be proved within the mathematical system from which they arose. The most famous example would be Riemann’s Hypothesis, involving primes. But the Goldbach conjecture (also involving primes) and the conjecture of twin primes also fit into this category. While most mathematicians believe them to be true, they are yet to be proven. I won’t elaborate on them, as they can easily be looked up.
 
But there is more: according to Gregory Chaitin, there are infinitely more incomputable Real numbers than computable Real numbers, which means that most of mathematics is inaccessible to logic.
 
So, when I say it all started with Euclid, I mean all the technology and infrastructure that we take for granted; and which allows me to write this so that virtually anyone anywhere in the world can read it; only exists because Euclid was able to derive ‘truths’ that stood for centuries and ultimately led to this.