Kant and modern physics

 I wrote a post on Kant back in February 2020, but it was actually an essay I wrote more than 20 years earlier, when I was a student of philosophy. I would not be able to improve on that essay, and I’m not about to try now. In that essay, I argue that Kant’s great contribution to philosophy, and epistemology in particular, was his idea of the ‘thing-in-itself’, which may remain forever unknowable, as we only have our perceptions of ‘things’.
 
In other posts, I have sometimes argued that the ‘thing-in-itself’ is dependent on the scale that we can observe it, but there is something deeper that I think only became apparent in the so-called golden age of physics in the 20th Century. In a more recent post, I pointed out that both relativity theory and quantum mechanics (the 2 pillars of modern physics) are both observer dependent. I argue that there could be an objective ontology that they can’t describe. I think this is more obvious in the case of special relativity, where different observers literally measure different durations of both space and time, but I’m getting ahead of myself.
 
On Quora, there are 4 physicists whom I ‘follow’ and read regularly. They are Viktor T Toth, Richard Muller, Mark John Fernee and Ian Miller. Out of these, Miller is possibly the most contentious as he argues against non-locality in QM (quantum mechanics), which I’m not aware of any other physicist concurring with. Of course, it’s Bell’s Inequality that provides the definitive answer to this, of which Miller has this to say:
 
If you say it must because of violations of Bell’s Inequality, first note that the inequality is a mathematical relationship that contains only numbers; no physical concept is included.
 
But the ‘numbers’ compare classical statistical outcomes with Born statistical outcomes and experiments verify Born’s results, so I disagree. Having said that, Miller makes pertinent points that I find insightful and, like all those mentioned, he knows a lot more about this topic than me.
 
For example, concerning relativity, he argues that it’s the ruler that changes dimension and not the space being measured. He also points out, regarding the twin paradox, that only one twin gains energy, which is the one whose clock slows down. Note that clocks are also a form of ‘ruler’, but they measure time instead of space. So you can have 2 observers who ‘measure’ different durations of space and time, but agree on ‘now’, when they reunite, as is the case with the twin paradox thought experiment.
 
This point is slightly off-track, but not irrelevant to the main focus of this post. The main focus is an academic paper jointly written by Shaun Maguire and Richard Muller, titled Now, and the Flow of Time. This paper is arguably as contentious as Miller’s take on non-locality and Bell, because Muller and Maguire argue that ‘space’ can be created.
 
Now, Viktor T Toth is quite adamant that space is not created because space is not an entity, but a ‘measurement’ between entities called ‘objects’. Now, it has to be said, that Muller has stated publicly on Quora that he has utmost respect for Toth and neither of them have called each other out over this issue.
 
Toth argues that people confound the mathematical metric with ‘space’ or ‘spacetime’, but I’d argue that this mathematical metric has physical consequences. In another post, I reference another paper, recommended to me by Mark John Fernee (authored by Tamara M. Davis and Charles H. Lineweaver at the University of New South Wales) which describes how a GR Doppler shift intrinsically measures the expansion of space.
 
The general relativistic interpretation of the expansion interprets cosmological redshifts as an indication of velocity since the proper distance between comoving objects increases. However, the velocity is due to the rate of expansion of space, not movement through space, and therefore cannot be calculated with the special relativistic Doppler shift formula. (My emphasis)
 
As I explain in that post: ‘What they are effectively saying is that there is a distinction between the movement of objects in space and the movement of space itself.’
 
The spacetime metric that Toth refers to provides a reference frame for c, the speed of light. So, whilst a spacetime metric (‘space’ by another name) can travel faster than light with respect to us (so over the horizon of the observable universe), an observer situated in that metric would still measure light as c relative to them.
 
Muller’s and Maguire’s paper goes even further, saying that space is created along with time, and they believe this can be measured as ‘a predicted lag in the emergence of gravitational radiation when two black holes merge.’ I won’t go into the details; you would need to read the paper.
 
A conclusion implicit in their theory is that there could be a universal now.
 
A natural question arises: why are the new nows created at the end of time, rather than uniformly throughout time, in the same way that new space is uniformly created throughout the universe.
 
The authors then provide alternative arguments, which I won’t go into, but they do ponder the fundamental difference between space and time, where one is uni-directional and the other is not. As far as we know, there is no ‘edge’ in space but there is in time. Muller and Maguire do wonder if space is ‘created’ throughout the Universe (as quoted above) or at an ‘edge’.
 
You may wonder how does Kant fit into all this? It’s because all these discussions are dependent on what we observe and what we theorise, both of which are perceptions. And, in physics, theorising involves mathematics. I’ve argued that mathematics can be seen as another medium determining perceptions, along with all the instruments we’ve built that now include the LHC and the Hubble and Webb telescopes.
 
Sabine Hossenfelder, whom I often reference on this blog these days, wrote a book, called Lost in Math, where she interviews some of the brightest minds in physics and challenges the pervading paradigm that mathematics can provide answers to questions that experimentation can’t – string theory being the most obvious.

Before the revolution in cosmology, created by Copernicus and built on by Galileo, Kepler and Newton, people believed that the Sun went round the Earth and that some objects in the night sky would occasionally backtrack in their orbits, which was explained by epicycles. That was overturned, and now it seems obvious that, in fact, the Earth rotates on its axis and orbits the sun along with all the other planets, which explains our ‘perception’ that sometimes the planets go ‘backwards.’
 
I wonder if the next revolution in science and cosmology may also provide a ‘simpler’ picture, where there is a ‘universal now’ that explains the age of the Universe, the edge of time that we all experience and non-locality in QM.
 
Of course, I’m probably wrong.

Addendum: This is Richard Muller talking about time on Quora.

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.

Ontology and epistemology; the twin pillars of philosophy

 I remember in my introduction to formal philosophy that there were 5 branches: ontology, epistemology, logic, aesthetics and ethics. Logic is arguably subsumed under mathematics, which has a connection with ontology and epistemology through physics, and ethics is part of all our lives, from politics to education to social and work-related relations to how one should individually live. Aesthetics is like an orphan in this company, yet art is imbued in all cultures in so many ways, it is unavoidable.
 
However, if you read about Western philosophy, the focus is often on epistemology and its close relation, if not utter dependence, on ontology. Why dependence? Because you can’t have knowledge of something without inferring its existence, even if the existence is purely abstract.
 
There are so many facets to this, that it’s difficult to know where to start, but I will start with Kant because he argued that we can never know ‘the-thing-in-itself’, only a perception of it, which, in a nutshell, is the difference between ontology and epistemology.
 
We need some definitions, and ontology is dictionary defined as the ‘nature of being’, while epistemology is ‘theory of knowledge’, and with these definitions, one can see straightaway the relationship, and Kant’s distillation of it.
 
Of course, one can also see how science becomes involved, because science, at its core, is an epistemological endeavour. In reading and researching this topic, I’ve come to the conclusion that, though science and philosophy have common origins in Western scholarship, going back to Plato, they’ve gone down different paths.
 
If one looks at the last century, which included the ‘golden age of physics’, in parallel with the dominant philosophical paradigm, heavily influenced, if not initiated, by Wittgenstein, we see that the difference can be definitively understood in terms of language. Wittgenstein effectively redefined epistemology as how we frame the world with language, while science, and physics in particular, frames the world in mathematics. I’ll return to this fundamental distinction later.
 
In my last post, I went to some lengths to argue that a fundamental assumption among scientists is that there is an ‘objective reality’. By this, I mean that they generally don’t believe in ‘idealism’ (like Donald Hoffman) which is the belief that objects don’t exist when you don’t perceive them (Hoffman describes it as the same experience as using virtual-reality goggles). As I’ve pointed out before, this is what we all experience when we dream, which I contend is different to the experience of our collective waking lives. It’s the word, ‘collective’, that is the key to understanding the difference – we share waking experiences in a way that is impossible to corroborate in a dream.
 
However, I’ve been reading a lot of posts on Quora by physicists, Viktor T Toth and Mark John Fernee (both of whom I’ve cited before and both of whom I have a lot of respect for). And they both point out that much of what we call reality is observer dependent, which makes me think of Kant.
 
Fernee, when discussing quantum mechanics (QM) keeps coming back to the ‘measurement problem’ and the role of the observer, and how it’s hard to avoid. He discusses the famous ‘Wigner’s friend’ thought experiment, which is an extension of the famous Schrodinger’s cat thought experiment, which infers you have the cat in 2 superpositional states: dead and alive. Eugne Wigner developed a thought experiment, whereby 2 experimenters could get contradictory results. Its relevance to this topic is that the ontology is completely dependent on the observer. My understanding of the scenario is that it subverts the distinction between QM and classical physics.
 
I’ve made the point before that a photon travelling across the Universe from some place and time closer to its beginning (like the CMBR) is always in the future of whatever it interacts with, like, for example, an ‘observer’ on Earth. The point I’d make is that billions of years of cosmological time have passed, so in another sense, the photon comes from the observer’s past, who became classical a long time ago. For the photon, time is always zero, but it links the past to the present across almost the entire lifetime of the observable universe.
 
Quantum mechanics, more than any other field, demonstrates the difference between ontology and epistemology, and this was discussed in another post by Fernee. Epistemologically, QM is described mathematically, and is so successful that we can ignore what it means ontologically. This has led to diverse interpretations from the multiple worlds interpretation (MWI) to so-called ‘hidden variables’ to the well known ‘Copenhagen interpretation’.
 
Fernee, in particular, discusses MWI, not that he’s an advocate, but because it represents an ontology that no one can actually observe. Both Toth and Fernee point out that the wave function, which arguably lies at the heart of QM is never observed and neither is its ‘decoherence’ (which is the measurement problem by another name), which leads many to contend that it’s a mathematical fiction. I argue that it exists in the future, and that only classical physics is actually observed. QM deals with probabilities, which is purely epistemological. After the ‘observation’, Schrodinger’s equation, which describes the wave function ceases to have any meaning. One is in the future and the observation becomes the past as soon as it happens.
 
I don’t know enough about it, but I think entanglement is the key to its ontology. Fernee points out in another post that entanglement is to do with conservation, whether it be the conservation of momentum or, more usually, the conservation of spin. It leads to what is called non-locality, according to Bell’s Theorem, which means it appears to break with relativistic physics. I say ‘appears’, because it’s well known that it can’t be used to send information faster than light; so, in reality, it doesn’t break relativity. Nevertheless, it led to Einstein’s famous quote about ‘spooky action at a distance’ (which is what non-locality means in layperson’s terms).
 
But entanglement is tied to the wave function decoherence, because that’s when it becomes manifest. It’s crucial to appreciate that entangled particles are described by the same wave function and that’s the inherent connection. It led Schrodinger to claim that entanglement is THE defining feature of QM; in effect, it’s what separates QM from classical physics.
 
I think QM is the best demonstration of Kant’s prescient claim that we can never know the-thing-in-itself, but only our perception of it. QM is a purely epistemological theory – the ontology it describes still eludes us.
 
But relativity theory also suggests that reality is observer dependent. Toth points out that even the number of particles that are detected in some scenarios are dependent on the frame of reference of the observer. This has led at least one physicist (on Quora) to argue that the word ‘particle’ should be banned from all physics text books – there are only fields. (Toth is an expert on QFT, quantum field theory, and argues that particles are a manifestation of QFT.) I won’t elaborate as I don’t really know enough, but what’s relevant to this topic is that time and space are observer dependent in relativity, or appear to be.
 
In a not-so-recent post, I described how different ‘observers’ could hypothetically ‘see’ the same event happening hundreds of years apart, just because they are walking across a street in opposite directions. I use quotation marks, because it’s all postulated mathematically, and, in fact, relativity theory prevents them from observing anything outside their past and future light cones. I actually discussed this with Fernee, and he pointed out that it’s to do with causality. Where there is no causal relation between events, we can’t determine an objective sequence let alone one relevant to a time frame independent of us (like a cosmic time frame). And this is where I personally have an issue, because, even though we can’t observe it or determine it, I argue that there is still an objective reality independently of us.
 
In relativity there is something called true time (τ) which is the time in the frame of reference of the observer. If spacetime is invariant, then it would logically follow that where you have true time you should have an analogous ‘true space’, yet I’ve never come across it. I also think there is a ‘true simultaneity’ but no one else does, so maybe I’m wrong.
 
There is, however, something called the Planck length, and someone asked Toth if this changed relativistically with the Lorenz transformation, like all other ‘rulers’ in relativity physics. He said that a version of relativity was formulated that made the Planck length invariant but it created problems and didn’t agree with experimental data. What I find interesting about this is that Planck’s constant, h, literally determines the size of atoms, and one doesn’t expect atoms to change size relativistically (but maybe they do). The point I’d make is that these changes are observer dependent, and I’d argue that there is a Planck length that is observer independent, which is the case when there is no observer.
 
This has become a longwinded way of explaining how 20th Century science has effectively taken this discussion away from philosophy, but it’s rarely acknowledged by philosophers, who take refuge in Wittgenstein’s conclusion that language effectively determines what we can understand of the world, because we think in a language and that limits what we can conceptualise. And he’s right, until we come up with new concepts requiring new language. Everything I’ve just discussed was completely unknown more than 120 years ago, for which we had no language, let alone concepts.
 
Some years ago, I reviewed a book by Don Cupitt titled, Above Us Only Sky, which was really about religion in a secular world. But, in it, Cupitt repeatedly argued that things only have meaning when they are ‘language-wrapped’ (his term) and I now realise that he was echoing Wittgenstein. However, there is a context in which language is magical, and that is when it creates a world inside your head, called a story.
 
I’ve been reading Bryan Magee’s The Great Philosophers, based on a series of podcasts with various academics in 1987, which started with Plato and ended with Wittgenstein. He discussed Plato with Myles Burnyeat, Professor of Ancient Philosophy at Oxford. Naturally, they discussed Socrates, the famous dialogues and the more famous Republic, but towards the end they turned to the Timaeus, which was a work on ‘mathematical science’, according to Burnyeat, that influenced Aristotle and Ptolemy.
 
It's worth quoting their last exchange verbatim:
 
Magee: For us in the twentieth century there is something peculiarly contemporary about the fact that, in the programme it puts forward for acquiring an understanding of the world, Plato’s philosophy gives a central role to mathematical physics.
 
Burnyeat: Yes. What Plato aspired to do, modern science has actually done. And so there is a sort of innate sympathy between the two which does not hold for Aristotle’s philosophy. (My emphasis)

Addendum: This is a very good exposition on the 'measurement problem' by Sabine Hossenfelder, which also provides a very good synopsis of the wave function (ψ), Schrodinger's equation and the Born rule.

How does science work?

 This post effectively piggybacks onto my last post, because, when it comes to knowledge and truth, nothing beats science except mathematics. It also coincides with me watching videos of Bryan Magee talking to philosophers, from 30 to 40 years ago and more. I also have a book with a collection of these ‘discussions’, so the ones I can’t view, I can read about. One gets an overall impression from these philosophers that, when it comes to understanding the philosophy of science, the last person you should ask is a scientist.
 
Now, I’m neither a scientist nor a proper philosopher, but it should be obvious to anyone who reads this blog that I’m interested in both. And where others see a dichotomy or a grudging disrespect, I see a marriage. There is one particular discussion that Magee has (with Hilary Putnam from Harvard, in 1977) that is headlined, The Philosophy of Science. Now, where Magee and his contemporaries turn to Kant, Hume and Descartes, I turn to Paul Davies, Roger Penrose and Richard Feynman, so the difference in perspective couldn’t be starker.
 
Where to start? Maybe I’ll start with a reference to my previous post by contending that what science excels in is explanation. In fact, one could define a scientific theory as an attempted explanation of a natural phenomenon, and science in general as the attempt to explain natural phenomena in all of their manifestations. This axiomatically rules out supernatural phenomena and requires that the natural phenomenon under investigation can be observed, either directly or indirectly, and increasingly with advanced technological instruments.
 
It's the use of the word ‘attempt’ that is the fly in the ointment, and requires elaboration. I use the word, attempt, because all theories, no matter how successful, are incomplete. This goes to the core of the issue and the heart of any debate concerning the philosophy of science, which hopefully becomes clearer as I progress.
 
But I’m going to start with what I believe are a couple of assumptions that science makes even before it gets going. One assumption is that there is an objective reality. This comes up if one discusses Hume, as Magee does with Professor John Passmore (from ANU). I don’t know when this took place, but it was before 1987 when the collection was published. Now, neither Magee nor Passmore are ‘idealists’ and they don’t believe Hume was either, but they iterate Hume’s claim that you can never know for certain that the world doesn’t exist when you’re not looking. Stephen Hawking also references this in his book, The Grand Design. In this context, idealism refers to a philosophical position that the world only exists as a consequence of minds (Donald Hoffman is the best known contemporary advocate). This is subtly different to ‘solipsism’, which is a condition we all experience when we dream, both of which I’ve discussed elsewhere.
 
There is an issue with idealism that is rarely discussed, at least from my limited exposure to the term, which is that everything must only exist in the present – there can be no history - if everything physically disappears when unobserved. And this creates a problem with our current knowledge of science and the Universe. We now know, though Hume wouldn’t have known, that we can literally see hundreds and even thousands of years into the past, just by looking at the night sky. In fact, using the technology I alluded to earlier, we can ‘observe’ the CMBR (cosmic microwave background radiation), so 380,000 years after the Big Bang (13.8 billion years ago). If there is no ‘objective reality’ then the science of cosmology makes no sense. I’m not sure how Hoffman reconciles that with his view, but he has similar problems with causality, which I’ll talk about next, because that’s the other assumption that I believe science makes.
 
This again rubs up against Hume, because it’s probably his most famous philosophical point that causality uses an inductive logic that can’t be confirmed. Just because 2 events happen sequentially, there is no way you can know that one caused the other. To quote Passmore in his conversation with Magee: “exactly how does past experience justify a conclusion about future behaviour?” In other words, using the example that Passmore does, just because you saw a rubber ball bounce yesterday, how can you be sure that it will do the same tomorrow? This is the very illustration of ‘inductive reasoning’.
 
To give another example that is often used to demonstrate this view in extremis, just because night has followed day in endless cycles for millennia, doesn’t guarantee it’s going to happen tomorrow. This is where science enters the picture because it can provide an explanation, which as I stated right at the beginning, is the whole raison d’etre of science. Night follows day as a consequence of the Earth rotating on its axis. In another post, written years ago, I discussed George Lakoff’s belief that all things philosophical and scientific can be understood as metaphor, so that the relationship between circular motion and periodicity is purely metaphorical. If one takes this to its logical conclusion, the literal everyday experience of night and day is just a metaphor.
 
But getting back to Hume’s scepticism, science shows that there is a causal relationship between the rotation of the Earth and our experience of night and day. This is a very prosaic example, but it demonstrates that the premise of causality lies at the heart of science. Remember, it’s only in the last 400 years or so that we discovered that the Earth rotates. This was the cause of Galileo’s fatally close encounter with the Inquisition, because it contradicted the Bible.
 
Now, some people, including Hoffman (he’s my default Devil’s advocate), argue that quantum mechanics (QM) rules out causality. I think Mark John Fernee (physicist with the University of Queensland) provides the best response by explaining how Born’s rule provides a mathematically expressed causal link between QM and classical physics. He argues, in effect, that it’s the ‘collapse’ of the wave function in QM that gives rise to the irreversibility in time between QM and classical physics (the so-called ‘measurement problem’) but is expressed as a probability by the Born rule, before the measurement or observation takes place. That’s longwinded and a little obtuse, but the ‘measurement’ turns a probability into an actual event – the transition from future to past (to paraphrase Freeman Dyson).
 
On the other hand, Hoffman argues that there is no causality in QM. To quote from the academic paper he cowrote with Chetan Prakash:
 
Our views on causality are consistent with interpretations of quantum theory that abandon microphysical causality… The burden of proof is surely on one who would abandon microphysical causation but still cling to macrophysical causation.
 
So Hoffman seems to think that there is a scientific consensus that causality does not arise in QM. But it’s an intrinsic part of the ‘measurement problem’, which is literally what is observed but eludes explanation. To quote Fernee:
 
While the Born rule looks to be ad hoc, it actually serves the function of ensuring that quantum mechanics obeys causality by ensuring that a quantum of action only acts locally (I can't actually think of any better way to state this). Therefore there really has to be a Born rule if causality is to hold.
 
Leaving QM aside, my standard response to this topic is somewhat blunt: if you don’t believe in causality, step in front of a bus (it’s a rhetorical device, not an instruction). Even Hoffman acknowledges in an online interview that he wouldn’t step in front of a train. I thought his argument specious because he compared it to taking an icon on a computer desktop (his go-to analogy) and putting it in the trash can. He exhorts us to take the train "seriously but not literally", just like a computer desktop icon (watch this video from 26.30 min).

That’s a lengthy detour, but causality is a such a core ‘belief’ in science that it couldn’t be ignored or glossed over.
 
Magee, in his discussion with Passmore, uses Einstein’s theory of gravity superseding Newton’s as an example of how a subsequent scientific theory can prove a previous theory ‘wrong’. In fact, Passmore compares it with the elimination of the ‘phlogiston’ theory by Lavoisier. But there is a dramatic difference. Phlogiston was a true or false theory in the same way that the Sun going around the Earth was a true or false theory, and, in both cases, they were proven ‘wrong’ by subsequent theories. That is not the case with Newton’s theory of gravitation.
 
It needs to be remembered that Newton’s theory was no less revolutionary than Einstein’s. He showed that the natural mechanism which causes (that word again) an object to fall to the ground on Earth is exactly the same mechanism that causes the moon to orbit the Earth. There is a reason why Newton is one of the few intellectual giants in history who is commonly compared with the more recent intellectual giant, Einstein.
 
My most pertinent point that I made right at the start is that all scientific theories are incomplete, and this applies to both Newton’s and Einstein’s theories of gravity. It’s just that Einstein’s theory is less incomplete than Newton’s and that is the real difference. And this is where I collide head-on with Magee and his interlocutors. They argue that the commonly held view that science progresses as a steady accumulation of knowledge is misleading, while I’d argue that the specific example they give – Einstein versus Newton – demonstrates that is exactly how science progresses, only it happens in quantum leaps rather than incrementally.
 
Thomas Kuhn wrote a seminal book, The Structure of Scientific Revolutions, which challenged the prevailing view that science progresses by incremental steps and this is the point that Magee is making. On this I agree: science has progressed by revolutions, yet it has still been built on what went before. As Claudia de Rahm (whom I wrote about in a former post) makes clear in a discussion on Einstein’s theory of gravity: any new theory that replaces it has to explain what the existing theory already explains. She specifically says, in answer to a question from her audience, that you don’t throw what we already know to be true (from empirical evidence) ‘into the rubbish bin’. And Einstein faced this same dilemma when he replaced Newton’s theory. In fact, one of his self-imposed criteria was that his theory must be mathematically equivalent to Newton’s when relativistic effects were negligible, which is true in most circumstances.
 
Passmore argues that Einstein’s theory even contradicts Newton’s theory, without being specific. The thing is that Einstein’s revolution affected the very bedrock of physics, being space and time. So maybe that’s what he’s referring to, because Newton’s theory assumed there was absolute space and absolute time, which Einstein effectively replaced with absolute spacetime.
 
I’ve discussed this in another post, but it bears repeating, because it highlights the issue in a way that is easily understood. Newton asks you to imagine a spinning bucket of water and observe what happens. And what happens is that the water surface becomes concave as a consequence of centrifugal forces. He then asked, what is it spinning in reference to? The answer is Earth, but the experiment applies to every spinning object in the Universe, including galaxies. They weren’t known in Newton’s time, nevertheless he had the insight to appreciate that the bucket spun relative to the stars in the night sky – in other words, with respect to the whole cosmos. Therefore, he concluded there must be absolute space, which is not spinning. Einstein, in answer to the same philosophical question, replaced absolute space with absolute spacetime.
 
In last week’s New Scientist (6 August 2022), Chanda Prescod-Weinstein (Assistant Professor in physics and astronomy at New Hampshire University) spent an entire page explaining how Einstein’s GR (General Theory of Relativity) is a ‘background independent theory’, which, in effect, means that it’s not dependent on a specific co-ordinate system. But within her discussion, she makes this point about the Newtonian perspective:
 
The theory [GR] did share something with the Newtonian perspective: while space and time were no longer absolute, they remained a stage on which events unfolded.
 
Another ‘truth’ that carries over from Newton to Einstein is the inverse square law, which has a causal relationship with planets, ensuring their orbits remain stable over astronomical time frames.
 
While Magee’s and Putnam’s discussion is ostensibly about the philosophy of science they mostly only talk about physics, which they acknowledge, and so have I. However, one should mention the theory of evolution (as they also do) because it demonstrates even better than the theory of gravitation, that science is a cumulative process. Everything we’ve learnt since Darwin’s and Wallace’s theory of natural selection has demonstrated that they were right, when it could have demonstrated they were wrong. And like Newton and Einstein, Darwin acknowledged the shortcomings in his theory – what he couldn’t explain.
 
But here’s the thing: in both cases, subsequent discoveries along with subsequent theories act like a filter, so what was true in a previous theory carries over and what was wrong is winnowed out. This is how I believe science works, which is distinct from Magee’s and Putnam’s account.
 
Putnam distinguishes between induction and deduction, pointing out that deduction can be done algorithmically on a computer while induction can’t. He emphasises at the start that induction along with empirical evidence is effectively the scientific method, but later he and Magee are almost dismissive of the scientific method, as if it’s past its use-by-date. This inference deserves closer analysis.
 
A dictionary definition of induction in this context is worth noting: the inference of a general law from particular instances. This is especially true in physics and has undoubtedly contributed to its success. Newton took the observation of an object falling on Earth and generalised it to include the entire solar system. He could only do this because of the work of Kepler who used the accurate observations of Tycho Brahe on the movements of the planets. Einstein then generalised the theory further, so that it was independent of any frame of reference or set of co-ordinates, as mentioned above.
 
The common thread that runs through all 3 of these iconoclasts (4 if you include Galileo) is mathematics. In fact, it was Galileo who famously said that if you want to read the book of nature, it is written in the language of mathematics (or words to that effect). A sentiment reiterated by Feynman (nearly 4 centuries later) in his book, The Character of Physical Law.
 
Einstein was arguably the first person who developed a theory based almost solely on mathematics before having it confirmed by observation, and a century later that has become such a common practice, it has led to a dilemma in physics. The reason that the scientific method is in crisis (if I can use that word) is because we can’t do the experiments to verify our theories, which is why the most ambitious theory in physics, string theory, has effectively stagnated for over a quarter of a century.
 
On the subject of mathematics and physics, Steven Weinberg was interviewed on Closer to Truth (posted last week), wherein he talks about the role of symmetry in elementary particle physics. It demonstrates how mathematics is intrinsic to physics at a fundamental level and integral to our comprehension.

 

Footnote: Sabine Hossenfelder, a theoretical physicist with her own YouTube channel (recommended) wrote a book, Lost in Math; How Beauty Leads Physics Astray (2018), where she effectively addresses the 'crisis' I refer to. In it, she interviews some of the smartest people in physics, including Steven Weinberg. She's also written her own book on philosophy, which is imminent. (Steven Weinberg passed away 23 July 2021)

What is knowledge? And is it true?

 This is the subject of a YouTube video I watched recently by Jade. I like Jade’s and Tibees’ videos, because they are both young Australian women (though Tibees is obviously a Kiwi, going by her accent) who produce science and maths videos, with their own unique slant. I’ve noticed that Jade’s videos have become more philosophical and Tibees’ often have an historical perspective. In this video by Jade, she also provides historical context. Both of them have taught me things I didn’t know, and this video is no exception.
 
The video has a different title to this post: The Gettier Problem or How do you know that you know what you know? The second title gets to the nub of it. Basically, she’s tackling a philosophical problem going back to Plato, which is how do you know that a belief is actually true? As I discussed in an earlier post, some people argue that you never do, but Jade discusses this in the context of AI and machine-learning.
 
She starts off with the example of using Google Translate to translate her English sentences into French, as she was in Paris at the time of making the video (she has a French husband, whom she’s revealed in other videos). She points out that the AI system doesn’t actually know the meaning of the words, and it doesn’t translate the way you or I would: by looking up individual words in a dictionary. No, the system is fed massive amounts of internet generated data and effectively learns statistically from repeated exposure to phrases and sentences so it doesn’t have to ‘understand’ what it actually means. Towards the end of the video, she gives the example of a computer being able to ‘compute’ and predict the movements of planets without applying Newton’s mathematical laws, simply based on historical data, albeit large amounts thereof.
 
Jade puts this into context by asking, how do you ‘know’ something is true as opposed to just being a belief? Plato provided a definition: Knowledge is true belief with an account or rational explanation. Jade called this ‘Justified True Belief’ and provides examples. But then, someone called Edmund Gettier mid last century demonstrated how one could hold a belief that is apparently true but still incorrect, because the assumed causal connection was wrong. Jade gives a few examples, but one was of someone mistaking a cloud of wasps for smoke and assuming there was a fire. In fact, there was a fire, but they didn’t see it and it had no connection with the cloud of wasps. So someone else, Alvin Goodman, suggested that a way out of a ‘Gettier problem’ was to look for a causal connection before claiming an event was true (watch the video).
 
I confess I’d never heard these arguments nor of the people involved, but I felt there was another perspective. And that perspective is an ‘explanation’, which is part of Plato’s definition. We know when we know something (to rephrase her original question) when we can explain it. Of course, that doesn’t mean that we do know it, but it’s what separates us from AI. Even when we get something wrong, we still feel the need to explain it, even if it’s only to ourselves.
 
If one looks at her original example, most of us can explain what a specific word means, and if we can’t, we look it up in a dictionary, and the AI translator can’t do that. Likewise, with the example of predicting planetary orbits, we can give an explanation, involving Newton’s gravitational constant (G) and the inverse square law.
 
Mathematical proofs provide an explanation for mathematical ‘truths’, which is why Godel’s Incompleteness Theorem upset the apple cart, so-to-speak. You can actually have mathematical truths without proofs, but, of course, you can’t be sure they’re true. Roger Penrose argues that Godel’s famous theorem is one of the things that distinguishes human intelligence from machine intelligence (read his Preface to The Emperor’s New Mind), but that is too much of a detour for this post.
 
The criterion that is used, both scientifically and legally, is evidence. Having some experience with legal contractual disputes, I know that documented evidence always wins in a court of law over undocumented evidence, which doesn’t necessarily mean that the person with the most documentation was actually right (nevertheless, I’ve always accepted the umpire’s decision, knowing I provided all the evidence at my disposal).
 
The point I’d make is that humans will always provide an explanation, even if they have it wrong, so it doesn’t necessarily make knowledge ‘true’, but it’s something that AI inherently can’t do. Best examples are scientific theories, which are effectively ‘explanations’ and yet they are never complete, in the same way that mathematics is never complete.
 
While on the topic of ‘truths’, one of my pet peeves are people who conflate moral and religious ‘truths’ with scientific and mathematical ‘truths’ (often on the above-mentioned basis that it’s impossible to know them all). But there is another aspect, and that is that so-called moral truths are dependent on social norms, as I’ve described elsewhere, and they’re also dependent on context, like whether one is living in peace or war.
 
Back to the questions heading this post, I’m not sure I’ve answered them. I’ve long argued that only mathematical truths are truly universal, and to the extent that such ‘truths’ determine the ‘rules’ of the Universe (for want of a better term), they also ultimately determine the limits of what we can know.

Creative and analytic thinking

I recently completed an online course with a similar title, How to Think Critically and Creatively. It must be the 8th or 9th course I’ve done through New Scientist, on a variety of topics, from cosmology and quantum mechanics to immunology and sustainable living; so quite diverse subjects. I started doing them during COVID, as they helped to pass the time and stimulate the brain at the same time.
 
All these courses rely on experts in their relevant fields from various parts of the globe, so not just UK based, as you might expect. This course was no exception with just 2 experts, both from America. Denise D Cummins is described as a ‘cognitive scientist, author and elected Fellow of the Association for Psychological Science, and she’s held faculty at Yale, UC, University of Illinois and the Centre of Adaptive Behaviours at the Max Planck Institute in Berlin’. Gerard J Puccio is ‘Department Chair and Professor at the International Centre for Studies on Creativity, Buffalo State; a unique academic department that offers the world’s only Master of Science degree in creativity’.
 
I admit to being sceptical that ‘creativity’ can be taught, but that depends on what one means by creativity. If creativity means using your imagination, then yes, I think it can, because imagination is something that we all have, and it’s probably a valid comment that we don’t make enough use of it in our everyday lives. If creativity means artistic endeavour then I think that’s another topic, even though it puts imagination centre stage, so to speak.
 
I grew up in a family where one side was obviously artistic and the other side wasn’t, which strongly suggests there’s a genetic component. The other side excelled at sport, and I was rubbish at sport. However, both sides were obviously intelligent, despite a notable lack of formal education; in my parents’ case, both leaving school in their early teens. In fact, my mother did most of her schooling by correspondence, and my father left school in the midst of the great depression, shortly followed by active duty in WW2.
 
Puccio (mentioned above) argues that creativity isn’t taught in our education system because it’s too hard. Instead, he says that we teach by memorising facts and by ‘understanding’ problems. I would suggest that there is a hierarchy, where you need some basics before you can ‘graduate’ to ‘creative thinking’, and I use the term here in the way he intends it. I spent most of my working lifetime on engineering projects, with diverse and often complex elements. I need to point out that I wasn’t one of the technical experts involved, but I worked with them, in all their variety, because my job was to effectively co-ordinate all their activities towards a common goal, by providing a plan and then keeping it on the rails.
 
Engineering is all about problem solving, and I’m not sure one can do that without being creative, as well as analytical. In fact, one could argue that there is a dialectical relationship between them, but maybe I’m getting ahead of myself.
 
Back to Puccio, who introduced 2 terms I hadn’t come across before: ‘divergent’ and ‘convergent’ thinking, arguing they should be done in that order. In a nutshell, divergent thinking is brainstorming where one thinks up as many options as possible, and convergent thinking is where one narrows in on the best solution. He argues that we tend to do the second one without doing the first one. But this is related to something else that was raised in the course, which is ‘Type 1 thinking’ and ‘Type 2 thinking’.
 
Type 1 thinking is what most of us would call ‘intuition’, because basically it’s taking a cognitive shortcut to arrive at an answer to a problem, which we all do all the time, especially when time is a premium. Type 2 thinking is when we analyse the problem, which is not only time consuming but takes up brain resources that we’d prefer not to use, because we’re basically lazy, and I’m no exception. These 2 cognitive behaviours are clinically established, so it’s not pop-science.
 
However, something that was not discussed in the course, is that type 2 thinking can become type 1 thinking when we develop expertise in something, like learning a musical instrument, or writing a story, or designing a building. In other words, we develop heuristics based on our experience, which is why we sometimes jump to convergent thinking without going through the divergent part.
 
The course also dealt with ‘critical thinking’, as per its title, but I won’t dwell on that, because critical thinking arises from being analytical, and separating true expertise from bogus expertise, which is really a separate topic.
 
How does one teach these skills? I’m not a teacher, so I’m probably not best qualified to say. But I have a lot of experience in a profession that requires analytical thinking and problem-solving as part of its job description. The one thing I’ve learned from my professional life is the more I’m restrained by ‘rules’, the worse job I’ll do. I require the freedom and trust to do things my own way, and I can’t really explain that, but it’s also what I provide to others. And maybe that’s what people mean by ‘creative thinking’; we break the rules.
 
Artistic endeavour is something different again, because it requires spontaneity. But there is ‘divergent thinking’ involved, as Puccio pointed out, giving the example of Hemingway writing countless endings to Farewell to Arms, before settling on the final version. I’m reminded of the reported difference between Beethoven and Mozart, two of the greatest composers in the history of Western classical music. Beethoven would try many different versions of something (in his head and on paper) before choosing what he considered the best. He was extraordinarily prolific but he wrote only 9 symphonies and 5 piano concertos plus one violin concerto, because he workshopped them to death. Mozart, on the other hand, apparently wrote down whatever came into his head and hardly revised it. One was very analytical in their approach and the other was almost completely spontaneous.
 
I write stories and the one area where I’ve changed type 2 thinking into type 1 thinking is in creating characters – I hardly give it a thought. A character comes into my head almost fully formed, as if I just met them in the street. Over time I learn more about them and they sometimes surprise me, which is always a good thing. I once compared writing dialogue to playing jazz, because they both require spontaneity and extemporisation. Don Burrows once said you can’t teach someone to play jazz, and I’ve argued that you can’t teach someone to write dialogue.
 
Having said that, I once taught a creative writing class, and I gave the class exercises where they were forced to write dialogue, without telling them that that was the point of the exercise. In other words, I got them to teach themselves.
 
The hard part of storytelling for me is the plot, because it’s a neverending exercise in problem-solving. How did I get back to here? Analytical thinking is very hard to avoid, at least for me.
 
As I mentioned earlier, I think there is a dialectic between analytical thinking and creativity, and the best examples are not artists but genii in physics. To look at just two: Einstein and Schrodinger, because they exemplify both. But what came first: the analysis or the creativity? Well, I’m not sure it matters, because they couldn’t have done one without the other. Einstein had an epiphany (one of many) where he realised that an object in free fall didn’t experience a force, which apparently contradicted Newton. Was that analysis or creativity or both? Anyway, he not only changed how we think about gravity, he changed the way we think about the entire cosmos.
 
Schrodinger, borrowed an idea from de Broglie that particles could behave like waves and changed how we think about quantum mechanics. As Richard Feynman once said, ‘No one knows where Schrodinger’s equation comes from. It came out of Schrodinger’s head. You can’t derive it from anything we know.’
 

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).

We are metaphysical animals

 I’m reading a book called Metaphysical Animals (How Four Women Brought Philosophy Back To Life). The four women were Mary Midgley, Iris Murdoch, Philippa Foot and Elizabeth Anscombe. The first two I’m acquainted with and the last two, not. They were all at Oxford during the War (WW2) at a time when women were barely tolerated in academia and had to be ‘chaperoned’ to attend lectures. Also a time when some women students ended up marrying their tutors. 

The book is authored by Clare Mac Cumhaill and Rachael Wiseman, both philosophy lecturers who became friends with Mary Midgley in her final years (Mary died in 2018, aged 99). The book is part biographical of all 4 women and part discussion of the philosophical ideas they explored.

 

Bringing ‘philosophy back to life’ is an allusion to the response (backlash is too strong a word) to the empiricism, logical positivism and general rejection of metaphysics that had taken hold of English philosophy, also known as analytical philosophy. Iris spent time in postwar Paris where she was heavily influenced by existentialism and Jean-Paul Sartre, in particular, whom she met and conversed with. 

 

If I was to categorise myself, I’m a combination of analytical philosopher and existentialist, which I suspect many would see as a contradiction. But this isn’t deliberate on my part – more a consequence of pursuing my interests, which are science on one hand (with a liberal dose of mathematical Platonism) and how-to-live a ‘good life’ (to paraphrase Aristotle) on the other.

 

Iris was intellectually seduced by Sartre’s exhortation: “Man is nothing else but that which he makes of himself”. But as her own love life fell apart along with all its inherent dreams and promises, she found putting Sartre’s implicit doctrine, of standing solitarily and independently of one’s milieu, difficult to do in practice. I’m not sure if Iris was already a budding novelist at this stage of her life, but anyone who writes fiction knows that this is what it’s all about: the protagonist sailing their lone ship on a sea full of icebergs and other vessels, all of which are outside their control. Life, like the best fiction, is an interaction between the individual and everyone else they meet. Your moral compass, in particular, is often tested. Existentialism can be seen as an attempt to arise above this, but most of us don’t. 

 

Not surprisingly, Wittgenstein looms large in many of the pages, and at least one of the women, Elizabeth Anscombe, had significant interaction with him. With Wittgenstein comes an emphasis on language, which has arguably determined the path of philosophy since. I’m not a scholar of Wittgenstein by any stretch of the imagination, but one thing he taught, or that people took from him, was that the meaning we give to words is a consequence of how they are used in ordinary discourse. Language requires a widespread consensus to actually work. It’s something we rarely think about but we all take for granted, otherwise there would be no social discourse or interaction at all. There is an assumption that when I write these words, they have the same meaning for you as they do for me, otherwise I am wasting my time.

 

But there is a way in which language is truly powerful, and I have done this myself. I can write a passage that creates a scene inside your mind complete with characters who interact and can cause you to laugh or cry, or pretty much any other emotion, as if you were present; as if you were in a dream.

 

There are a couple of specific examples in the book which illustrate Wittgenstein’s influence on Elizabeth and how she used them in debate. They are both topics I have discussed myself without knowing of these previous discourses.

 

In 1947, so just after the war, Elizabeth presented a paper to the Cambridge Moral Sciences Club, which she began with the following disclosure:

 

Everywhere in this paper I have imitated Dr Wittgenstein’s ideas and methods of discussion. The best that I have written is a weak copy of some features of the original, and its value depends only on my capacity to understand and use Dr Wittgenstein’s work.

 

The subject of her talk was whether one can truly talk about the past, which goes back to the pre-Socratic philosopher, Parmenides. In her own words, paraphrasing Parmenides, ‘To speak of something past’ would then to ‘point our thought’ at ‘something there’, but out of reach. Bringing Wittgenstein into the discussion, she claimed that Parmenides specific paradox about the past arose ‘from the way that thought and language connect to the world’.

 

We apply language to objects by naming them, but, in the case of the past, the objects no longer exist. She attempts to resolve this epistemological dilemma by discussing the nature of time as we experience it, which is like a series of pictures that move on a timeline while we stay in the present. This is analogous to my analysis that everything we observe becomes the past as soon as it happens, which is exemplified every time someone takes a photo, but we remain in the present – the time for us is always ‘now’.

 

She explains that the past is a collective recollection, documented in documents and photos, so it’s dependent on a shared memory. I would say that this is what separates our recollection of a real event from a dream, which is solipsistic and not shared with anyone else. But it doesn’t explain why the past appears fixed and the future unknown, which she also attempted to address. But I don’t think this can be addressed without discussing physics.

 

Most physicists will tell you that the asymmetry between the past and future can only be explained by the second law of thermodynamics, but I disagree. I think it is described, if not explained, by quantum mechanics (QM) where the future is probabilistic with an infinitude of possible paths and classical physics is a probability of ONE because it’s already happened and been ‘observed’. In QM, the wave function that gives the probabilities and superpositional states is NEVER observed. The alternative is that all the futures are realised in alternative universes. Of course, Elizabeth Anscombe would know nothing of these conjectures.

 

But I would make the point that language alone does not resolve this. Language can only describe these paradoxes and dilemmas but not explain them.

 

Of course, there is a psychological perspective to this, which many people claim, including physicists, gives the only sense of time passing. According to them, it’s fixed: past, present and future; and our minds create this distinction. I think our minds create the distinction because only consciousness creates a reference point for the present. Everything non-sentient is in a causal relationship that doesn’t sense time. Photons of light, for example, exist in zero time, yet they determine causality. Only light separates everything in time as well as space. I’ve gone off-topic.

 

Elizabeth touched on the psychological aspect, possibly unintentionally (I’ve never read her paper, so I could be wrong) that our memories of the past are actually imagined. We use the same part of the brain to imagine the past as we do to imagine the future, but again, Elizabeth wouldn’t have known this. Nevertheless, she understood that our (only) knowledge of the past is a thought that we turn into language in order to describe it.

 

The other point I wish to discuss is a famous debate she had with C.S. Lewis. This is quite something, because back then, C.S. Lewis was a formidable intellectual figure. Elizabeth’s challenge was all the more remarkable because Lewis’s argument appeared on the surface to be very sound. Lewis argued that the ‘naturalist’ position was self-refuting if it was dependent on ‘reason’, because reason by definition (not his terminology) is based on the premise of cause and effect and human reason has no cause. That’s a simplification, nevertheless it’s the gist of it. Elizabeth’s retort:

 

What I shall discuss is this argument’s central claim that a belief in the validity of reason is inconsistent with the idea that human thought can be fully explained as the product of non-rational causes.

 

In effect, she argued that reason is what humans do perfectly naturally, even if the underlying ‘cause’ is unknown. Not knowing the cause does not make the reasoning irrational nor unnatural. Elizabeth specifically cited the language that Lewis used. She accused him of confusing the concepts of “reason”, “cause” and “explanation”.

 

My argument would be subtly different. For a start, I would contend that by ‘reason’, he meant ‘logic’, because drawing conclusions based on cause and effect is logic, even if the causal relations (under consideration) are assumed or implied rather than observed. And here I contend that logic is not a ‘thing’ – it’s not an entity; it’s an action - something we do. In the modern age, machines perform logic; sometimes better than we do.

 

Secondly, I would ask Lewis, does he think reason only happens in humans and not other animals? I would contend that animals also use logic, though without language. I imagine they’d visualise their logic rather than express it in vocal calls. The difference with humans is that we can perform logic at a whole different level, but the underpinnings in our brains are surely the same. Elizabeth was right: not knowing its physical origins does not make it irrational; they are separate issues.

 

Elizabeth had a strong connection to Wittgenstein right up to his death. She worked with him on a translation and edit of Philosophical Investigations, and he bequeathed her a third of his estate and a third of his copyright.

 

It’s apparent from Iris’s diaries and other sources that Elizabeth and Iris fell in love at one point in their friendship, which caused them both a lot of angst and guilt because of their Catholicism. Despite marrying, Iris later had an affair with Pip (Philippa).

 

Despite my discussion of just 2 of Elizabeth’s arguments, I don’t have the level of erudition necessary to address most of the topics that these 4 philosophers published in. Just reading the 4 page Afterwards, it’s clear that I haven’t even brushed the surface of what they achieved. Nevertheless, I have a philosophical perspective that I think finds some resonance with their mutual ideas. 

 

I’ve consistently contended that the starting point for my philosophy is that for each of us individually, there is an inner and outer world. It even dictates the way I approach fiction. 

 

In the latest issue of Philosophy Now (Issue 149, April/May 2022), Richard Oxenberg, who teaches philosophy at Endicott College in Beverly, Massachusetts, wrote an article titled, What Is Truth? wherein he describes an interaction between 2 people, but only from a purely biological and mechanical perspective, and asks, ‘What is missing?’ Well, even though he doesn’t spell it out, what is missing is the emotional aspect. Our inner world is dominated by emotional content and one suspects that this is not unique to humans. I’m pretty sure that other creatures feel emotions like fear, affection and attachment. What’s more I contend that this is what separates, not just us, but the majority of the animal kingdom, from artificial intelligence.

 

But humans are unique, even among other creatures, in our ability to create an inner world every bit as rich as the one we inhabit. And this creates a dichotomy that is reflected in our division of arts and science. There is a passage on page 230 (where the authors discuss R.G. Collingwood’s influence on Mary), and provide an unexpected definition.

 

Poetry, art, religion, history, literature and comedy are all metaphysical tools. They are how metaphysical animals explore, discover and describe what is real (and beautiful and good). (My emphasis.)

 

I thought this summed up what they mean with their coinage, metaphysical animals, which titles the book, and arguably describes humanity’s most unique quality. Descriptions of metaphysics vary and elude precise definition but the word, ‘transcendent’, comes to mind. By which I mean it’s knowledge or experience that transcends the physical world and is most evident in art, music and storytelling, but also includes mathematics in my Platonic worldview.

 

Footnote: I should point out that certain chapters in the book give considerable emphasis to moral philosophy, which I haven’t even touched on, so another reader might well discuss other perspectives.

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 argues that all the non-trivial zeros of the Riemann Zeta function lie on a line in the complex plane which is 1/2i. In other words, all the zeros are of the form, a + 1/2i, which is a complex number with imaginary 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.