I recently wrote a post titled Kant and modern physics, plus I’d written a much more extensive essay on Kant previously, as well as an essay on Plato, whose famous Academy was arguably the origin of Western philosophy, science and mathematics.
This is in answer to a question on Quora. The first thing I did was turn the question inside out or upside down, as I explain in the opening paragraph. It was upvoted by Kip Wheeler, who describes himself as “Been teaching medieval stuff at Uni since 1993.” He provided his own answer to the same question, giving a contrary response to mine, so I thought his upvote very generous.
There are actually a lot of answers on Quora addressing this theme, and I only reference one of them. But, as far as I can tell, I’m the only one who links Plato to Kant to modern physics.
Why could Plato's theory of forms not help us to know things better?
I think this question is back-to-front. If you change ‘could’ to ‘would’ and eliminate ‘not’, the question makes more sense – at least, to me. Nevertheless, it ‘could… not help us to know things better’ if it’s misconstrued or if it’s merely considered a religious artefact with no relevance to contemporary epistemology.
There are some good answers to similar questions, with Paul Robinson’s answer to Is Plato’s “Theory of Ideas” True? being among the more erudite and scholarly. I won’t attempt to emulate him, but take a different tack using a different starting point, which is more widely known.
Robinson, among others, makes reference to Plato’s famous shadows on the wall of a cave allegory (or analogy in modern parlance), and that’s a good place to start. Basically, the shadows represent our perceptions of reality whilst ‘true’ reality remains unknown to us. Plato believed that there was a world of ‘forms’, which were perfect compared to the imperfect world we inhabit. This is similar to the Christian idea of Heaven as distinct from Earth, hence the religious connotation, which is still referenced today.
But there is another way to look at this, which is closer to Kant’s idea of the thing-in-itself. Basically, we may never know the true nature of something just based on our perceptions, and I’d contend that modern science, especially physics, has proved Kant correct, specifically in ways he couldn’t foresee.
That’s partly because we now have instruments and technologies that can change what we can perceive at all scales, from the cosmological to the infinitesimal. But there’s another development which has happened apace and contributed to both the technology and the perception in a self-reinforcing dialectic between theory and observation. I’m talking about physics, which is arguably the epitome of epistemological endeavour.
And the key to physics is mathematics, only there appears to be more mathematics than we need. Ever since the Scientific Revolution, mathematics has proven fundamental in our quest for the elusive thing-in-itself. And this has resulted in a resurgence in the idea of a Platonic realm, only now it’s exclusive to mathematics. I expect Plato would approve, since his famous Academy was based on Pythagoras’s quadrivium of arithmetic, geometry, astronomy and music, all of which involve mathematics.
16 April 2023
From Plato to Kant to physics
18 February 2024
What would Kant say?
Even though this is a philosophy blog, my knowledge of Western philosophy is far from comprehensive. I’ve read some of the classic texts, like Aristotle’s Nicomachean Ethics, Descartes Meditations, Hume’s A treatise of Human Nature, Kant’s Critique of Pure Reason; all a long time ago. I’ve read extracts from Plato, as well as Sartre’s Existentialism is a Humanism and Mill’s Utilitarianism. As you can imagine, I only recollect fragments, since I haven’t revisited them in years.
Nevertheless, there are a few essays on this blog that go back to the time when I did. One of those is an essay on Kant, which I retitled, Is Kant relevant to the modern world? Not so long ago, I wrote a post that proposed Kant as an unwitting bridge between Plato and modern physics. I say, ‘unwitting’, because, as far as I know, Kant never referenced a connection to Plato, and it’s quite possible that I’m the only person who has. Basically, I contend that the Platonic realm, which is still alive and well in mathematics, is a good candidate for Kant’s transcendental idealism, while acknowledging Kant meant something else. Specifically, Kant argued that time and space, like sensory experiences of colour, taste and sound, only exist in the mind.
Here is a good video, which explains Kant’s viewpoint better than me. If you watch it to the end, you’ll find the guy who plays Devil’s advocate to the guy expounding on Kant’s views makes the most compelling arguments (they’re both animated icons).
But there’s a couple of points they don’t make which I do. We ‘sense’ time and space in the same way we sense light, sound and smell to create a model inside our heads that attempts to match the world outside our heads, so we can interact with it without getting killed. In fact, our modelling of time and space is arguably more important than any other aspect of it.
I’ve always had a mixed, even contradictory, appreciation of Kant. I consider his insight that we may never know the things-in-themselves to be his greatest contribution to epistemology, and was arguably affirmed by 20th Century physics. Both relativity and quantum mechanics (QM) have demonstrated that what we observe does not necessarily reflect reality. Specifically, different observers can see and even measure different parameters of the same event. This is especially true when relativistic effects come into play.
In relativity, different observers not only disagree on time and space durations, but they can’t agree on simultaneity. As the Kant advocate in the video points out, surely this is evidence that space and time only exist in the mind, as Kant originally proposed. The Devil’s advocate resorts to an argument of 'continuity', meaning that without time as a property independent of the mind, objects and phenomena (like a candle burning) couldn’t continue to happen without an observer present.
But I would argue that Einstein’s general theory of relativity, which tells us that different observers can measure different durations of space and time (I’ll come back to this later), also tells us that the entire universe requires a framework of space and time for the objects to exist at all. In other words, GR tells us, mathematically, that there is an interdependence between the gravitational field that permeates and determines the motion of objects throughout the entire universe, and the spacetime metric those same objects inhabit. In fact, they are literally on opposite sides of the same equation.
And this brings me to the other point that I think is missing in the video’s discussion. Towards the end, the Devil’s advocate introduces ‘the veil of perception’ and argues:
We can only perceive the world indirectly; we have no idea what the world is beyond this veil… How can we then theorise about the world beyond our perceptions? …Kant basically claims that things-in-themselves exist but we do not know and cannot know anything about these things-in-themselves… This far-reaching world starts to feel like a fantasy.
But every physicist has an answer to this, because 20th Century physics has taken us further into this so-called ‘fantasy’ than Kant could possibly have imagined, even though it appears to be a neverending endeavour. And it’s specifically mathematics that has provided the means, which the 2 Socratic-dialogue icons have ignored. Which is why I contend that it’s mathematical Platonism that has replaced Kant’s transcendental idealism. It’s rendered by the mind yet it models reality better than anything else we have available. It’s the only means we have available to take us behind ‘the veil of perception’ and reveal the things-in-themselves.
And this leads me to a related point that was actually the trigger for me writing this in the first place.
In my last post, I mentioned I’m currently reading Kip A. Thorne’s book, Black Holes and Time Warps; Einstein’s Outrageous Legacy (1994). It’s an excellent book on many levels, because it not only gives a comprehensive history, involving both Western and Soviet science, it also provides insights and explanations most of us are unfamiliar with.
To give an example that’s relevant to this post, Thorne explains how making measurements at the extreme curvature of spacetime near the event horizon of a black hole, gives the exact same answer whether it’s the spacetime that distorts while the ‘rulers’ remain unchanged, or it’s the rulers that change while it’s the spacetime that remains ‘flat’. We can’t tell the difference. And this effectively confirms Kant’s thesis that we can never know the things-in-themselves.
To quote Thorne:
What is the genuine truth? Is spacetime really flat, or is it really curved? To a physicist like me this is an uninteresting question because it has no physical consequences (my emphasis). Both viewpoints, curved spacetime and flat, give the same predictions for any measurements performed with perfect rulers and clocks… (Earlier he defines ‘perfect rulers and clocks’ as being derived at the atomic scale)
Ian Miller (a physicist who used to be active on Quora) once made the point, regarding space-contraction, that it’s the ruler that deforms and not the space. And I’ve made the point myself that a clock can effectively be a ruler, because a clock that runs slower measures a shorter distance for a given velocity, compared to another so-called stationary observer who will measure the same distance as longer. This happens in the twin paradox thought experiment, though it’s rarely mentioned (even by me).
24 February 2020
Is Kant relevant to the modern world?
‘All these faculties have a transcendental (as well as an empirical) employment which concerns the form alone, and is possible apriori.’ By ‘apriori’ and ‘form’, Kant of course is referring to space and time, but he is also referring to mathematical forms, as he explains on the next page in B128. There is then, this relationship between transcendental idealism and empirical realism; a relationship that is mediated principally through mathematics.
14 November 2022
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.
07 September 2022
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 (Ï„) 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.
07 April 2024
What does physics really tell us about reality?
A little while ago I got into another discussion with Mark John Fernee (see previous post), but this time dealing with the relationship between ontology and epistemology as determined by physics. It came about in reference to a paper in Physics Today that someone cited, by N. David Nermin, a retired Professor of physics in Ithaca, New York, titled What’s bad about this habit. In particular, he talked about our tendency to ‘reify’ mathematically determined theories into reality. It helps if we have some definitions, which Fernee conveniently provided that were both succinct and precise.
Epistemology - concerning knowledge.
Ontology - concerning reality.
Reify - to think of an idea as real.
It so happens that around the same time I read an article in New Scientist (25 Mar 2024, pp.32-5) Strange but true? by philosopher, Eric Schwitzgebel, which covers similar territory. The title tells you little, but it’s really about how modern theories in physics don’t really tell us what reality is; instead giving us a range of possibilities to choose from.
I will start with Nermin, who spends the first page talking about quantum mechanics (QM), as it’s the most obvious candidate for a mathematical theory that gets reified by almost everyone who encounters it. This selected quote gives a good feel for what he’s talking about.
When I was a graduate student learning quantum field theory, I had a friend who was enchanted by the revelation that quantum fields were the real stuff that makes up the world. He reified quantum fields. But I hope you will agree that you are not a continuous field of operators on an infinite-dimensional Hilbert space. Nor, for that matter, is the page you are reading or the chair you are sitting in. Quantum fields are useful mathematical tools. They enable us to calculate things.
I found another quote by Freeman Dyson (2014), who makes a similar point to Nermin about the wave function (Ψ).
Unfortunately, people writing about quantum mechanics often use the phrase "collapse of the wave-function" to describe what happens when an object is observed. This phrase gives a misleading idea that the wave-function itself is a physical object. A physical object can collapse when it bumps into an obstacle. But a wave-function cannot be a physical object. A wave-function is a description of a probability, and a probability is a statement of ignorance. Ignorance is not a physical object, and neither is a wave-function. When new knowledge displaces ignorance, the wave-function does not collapse; it merely becomes irrelevant.
But Nermin goes on to challenge even the reality of space and time. Arguing that it is a mathematical abstraction.
What about spacetime itself? Is that real? Spacetime is a (3+1) dimensional mathematical continuum. Even if you are a mathematical Platonist, I would urge you to consider that this continuum is nothing more than an extremely effective way to represent relations between distinct events.
He then goes on to explain that ‘an event… can be represented as a mathematical point in spacetime.’
He elaborates how this has become so reified into ordinary language that we’re no longer aware that it is an abstraction.
So spacetime is an abstract four-dimensional mathematical continuum of points that approximately represent phenomena whose spatial and temporal extension we find it useful or necessary to ignore. The device of spacetime has been so powerful that we often reify that abstract bookkeeping structure, saying that we inhabit a world that is such a four (or, for some of us, ten) dimensional continuum. The reification of abstract time and space is built into the very languages we speak, making it easy to miss the intellectual sleight of hand.
And this is where I start to have issues with his overall thesis, whereas Fernee said, ‘I completely concur with what he has written, and it is well articulated.’
When I challenged Fernee specifically on Nermin’s points about space-time, Fernee argued:
His contention was that even events in space-time are an abstraction. We all assume the existence of an objective reality, and I don't know of anyone who would seriously challenge that idea. Yet our descriptions are abstractions. All we ask of them is that they are consistent, describe the observed phenomena, and can be used to make predictions.
I would make an interesting observation on this very point, that distinguishes an AI’s apparent perspective of space and time compared to ours. Even using the word, ‘apparent’, infers there is a difference that we don’t think about.
I’ve made the point in other posts, including one on Kant, that we create a model of space and time in our heads which we use to interact with the physical space and time that exists outside our heads, and so do all living creatures with eyes, ears and touch. In fact, the model is so realistic that we think it is the external reality.
When we throw or catch a ball on the sporting field, we know that our brains don’t work out the quadratic equations that determine where it’s going to land. But imagine an AI controlled artillery device, which would make those calculations and use a 3-dimensional grid to determine where its ordinance was going to hit. Likewise, imagine an AI controlled drone using GPS co-ordinates – in other words, a mathematical abstraction of space and time – to navigate its way to a target. And that demonstrates the fundamental difference that I think Nermin is trying to delineate. The point is that, from our perspective, there is no difference.
This quote gives a clearer description of Nermin’s philosophical point of view.
Space and time and spacetime are not properties of the world we live in but concepts we have invented to help us organize classical events. Notions like dimension or interval, or curvature or geodesics, are properties not of the world we live in but of the abstract geometric constructions we have invented to help us organize events. As Einstein once again put it, “Space and time are modes by which we think, not conditions under which we live.”
Whereas I’d argue that they are both, and the mathematics tells us things about the ‘properties of the world [universe]’ which we can’t directly perceive with our senses – like ‘geodesics’ and the ‘curvature’ of spacetime. Yet they can be measured as well as calculated, which is why we know GR (Einstein’s general theory of relativity) works.
My approach to understanding physics, which may be misguided and would definitely be the wrong approach according to Nermin and Fernee, is to try and visualise the concepts that the maths describes. The concept of a geodesic is a good example. I’ve elaborated on this in another post, but I can remember doing Newtonian-based physics in high school, where gravity made no sense to me. I couldn’t understand why the force of gravity seemed to be self-adjusting so that the acceleration (g) was the same for all objects, irrespective of their mass.
It was only many years later, when I understood the concept of a geodesic using the principle of least action, that it all made sense. The objects don’t experience a force per se, but travel along the path of least action which is also the path of maximum relativistic time. (I’ve described this phenomenon elsewhere.) The point is that, in GR, mass is not in the equations (unlike Newton’s mathematical representation) and the force we all experience is from whatever it is that stops us falling, which could be a chair you’re sitting on or the Earth.
So, the abstract ‘geodesic’ explains what Newton couldn’t, even though Newton gave us the right answers for most purposes.
And this leads me to extend the discussion to include the New Scientist article. The author, Eric Schwitzgebel, ponders 3 areas of scientific inquiry: quantum mechanics (are there many worlds?); consciousness (is it innate in all matter?) and computer simulations (do we live in one?). I’ll address them in reverse order, because that’s easiest.
As Paul Davies pointed out in The Goldilocks Enigma, the so-called computer-simulation hypothesis is a variant on Intelligent Design. If you don’t believe in ID, you shouldn’t believe that the universe is a computer simulation, because some entity had to design it and produce the code.
'Is consciousness innate?' is the same as pansychism, as Schwitzgebel concurs, and I’d say there is no evidence for it, so not worth arguing about. Basically, I don’t want to waste time on these 2 questions, and, to be fair, Schwitzgebel’s not saying he’s an advocate for either of them.
Which brings me to QM, and that’s relevant. Schwitzbegel makes the point that all the scientific interpretations have bizarre or non-common-sensical qualities, of which MWI (many worlds interpretation) is just one. Its relevance to this discussion is that they are all reifications that are independent of the mathematics, because the mathematics doesn’t discern between them. And this gets to the nub of the issue for me. Most physicists would agree that physics, in a nutshell, is about creating mathematical models that are then tested by experimentation and observation, often using extremely high-tech, not-to-mention behemoth instruments, like the LHC and the James Webb telescope.
It needs to be pointed out that, without exception, all these mathematical models have limitations and, historically, some have led us astray. The most obvious being Ptolemy’s model of the solar system involving epicycles. String theory, with its 10 dimensions and 10^500 possible universes, is a potential modern-day contender, but we don’t really know.
Nevertheless, as I explained with my brief discourse on geodesics (above), there are occasions when the mathematics provides an insight we would otherwise be ignorant of.
Basically, I think what Schwitzgebel is really touching on is the boundary between philosophy and science, which I believe has always existed and is an essential dynamic, despite the fact that many scientists are dismissive of its role.
Returning to Nermin, it’s worth quoting his final passage.
Quantum mechanics has brought home to us the necessity of separating that irreducibly real experience from the remarkable, beautiful, and highly abstract super-structure we have found to tie it all together.
The ‘real experience’ includes the flow of time; the universality of now which requires memory for us to know it exists; the subjective experience of free will. All of these are considered ‘illusions’ by many scientists, not least Sabine Hossenfelder in her excellent book, Existential Physics. I tend to agree with another physicist, Richard Muller, that what this tells us is that there is a problem with our theories and not our reality.
In an attempt to reify QM with reality, I like the notion proposed by Freeman Dyson that it’s a mathematical model that describes the future. As he points out, it gives us probabilities, and it provides a logical reason why Feynman’s abstraction of an infinite number of ‘paths’ are never observed.
Curiously, Fernee provides tacit support for the idea that the so-called ‘measurement’ or ‘observation’ provides an ‘abstract’ distinction between past and future in physics, though he doesn’t use those specific words.
In quantum mechanics, the measurement hypothesis, which includes the collapse of the wave function, is an irreversible process. As we perceive the world through measurements, time will naturally seem irreversible to us.
Very similar to something Davies said in another context:
The very act of measurement breaks the time symmetry of quantum mechanics in a process sometimes described as the collapse of the wave function…. the rewind button is destroyed as soon as that measurement is made.
Lastly, I would like to mention magnetism, because, according to SR, it’s mathematically dependent on a moving electric charge. Only it’s not always, as this video explicates. You can get a magnetic field from electric spin, which is an abstraction, as no one suggests that electrons do physically spin, even though they produce measurable magnetic moments.
What most people don’t know is that our most common experience of a magnetic field, which is a bar magnet, is created purely by electron spin and not moving electrons.
04 February 2012
Is mathematics invented or discovered?
I would argue that it is a mixture of both, in the same way that our scientific investigations are a combination of inventiveness and discovery. The difference is, that in science, the roles of creativity and discovery are more clearly delineated. We create theories, hypotheses and paradigms, and we perform experiments to observe results, and we also, sometimes, simply perform observations without a hypothesis and make discoveries, though this wouldn’t necessarily be considered scientific.
But there is a link between science and mathematics, because as our knowledge and investigations go deeper into uncovering nature’s secrets, we become more dependent on mathematics. In fact I would contend that the limit of our knowledge in science is determined by the limits of our mathematical abilities. It is only our ability to uncover complex and esoteric mathematical laws that has allowed us to uncover the most esoteric (some would say spooky) aspects of the natural universe. To the physicist there appears to be a link between mathematical laws and natural laws. Roger Penrose made the comment in a BBC programme, Lords of Time, to paraphrase him, that mathematics exists in nature. It is a sentiment that I would concur with. But to many philosophers, this link is an illusion of our own making.
Stanilas Debaene, in his book, The Number Sense, describes the cognitive aspect of our numeracy skills which can be found in pre-language infants as well as many animals. He argues a case that numbers, the basic building blocks of all mathematics, are created in our minds and that there is no such thing as natural numbers. The logical consequence of this argument is that if numbers are a product of the mind then so must be the whole edifice of mathematics. This is in agreement with both Russell and Wittgenstein, who are the most dominant figures in 20th Century philosophy. I have no problem with the notion that numbers exist only as a concept in the human mind, and that they even exist within the minds of some animals up to about 5 (if one reads Debaene’s book) though of course the animals aren’t aware that they have concepts – it’s just that they can count to a rudimentary level.
But mathematics, as we practice it, is not so much about numbers as the relationships that exist between numbers, which follow very precise rules and laws. In fact, the great beauty of algebra is that it strips mathematics of its numbers so that we can merely see the relationships. I have always maintained that mathematical rules are, by and large, not man made, and in fact are universal. From this perspective, Mathematics is a universal language, and it is the ideal tool for uncovering nature’s secrets because nature also obeys mathematical rules and laws. The modern philosopher argues that mathematics is merely logic, created by the human mind, albeit a very complex logic, from which we create models to approximate nature. This is a very persuasive argument, but do we bend mathematics to approximate nature, or is mathematics an inherent aspect of nature that allows an intelligence like ours to comprehend it?
I would argue that relationships like Ï€ and Pythagoras’s triangle, and the differential and integral calculus are discovered, not invented. We simply invent the symbols and the means to present them in a comprehensible form for our minds. If you have a problem and you cannot find the solution, does that mean the solution does not exist? Does the solution only exist when someone has unravelled it, like Fermat’s theorem? This is a bit like Schrodinger’s cat; it’s only dead or alive when someone has made an observation. So mathematical theorems and laws only exist when a cognitive mind somewhere reveals them. But do they also exist in nature like Bernoulli’s spiral found in the structure of a shell or a spider web, or Einstein’s equations describing the curvature of space? The modern philosopher would say Einstein’s equations are only an approximation, and he or she may be right, because nature has this habit of changing its laws depending on what scale we observe it at (see Addendum below), which leads paradoxically to the apparent incompatibility of Einstein’s equations with quantum mechanics. This is not unlike the mathematical conundrum of a circle, ellipse, parabola and hyperbola describing different aspects of a curve.
So what we have is this connection between the human mind and the natural world bridged by mathematics. Is mathematics an invention of the mind, a phenomenon of the natural world, or a confluence of both? I would argue that it is the last. Mathematics allows us to render nature’s laws in a coherent and accurate structure – it has the same infinite flexibility while maintaining a rigid consistency. This reads like a contradiction until you take into account two things. One is that nature is comprised of worlds within worlds, each one self-consistent but producing different entities at different levels. The best example is the biological cells that comprise the human body compared to the molecules that makes up the cells, and then in comparison with an individual human, the innumerable social entities that a number of humans can create. Secondly, that this level of complexity appears to be never ending so that our discoveries have infinite potential. This is despite the fact that in every age of technological discovery and invention, we have always believed that we almost know everything that there is to know. The current age is no different in this respect.
The philosophical viewpoint that I prescribe to does not require a belief in the Platonic realm. From my point of view, I consider it to be more Pythagorean than Platonic, because my understanding is that Pythagoras saw mathematics in nature in much the same way that Penrose expresses it. I assume this view, even though we have little direct knowledge of Pythagoras’s teachings. Plato, on the other hand, prescribed an idealised world of forms. He believed that because we’ve had previous incarnations (an idea he picked up from Pythagoras, who was a religious teacher first, mathematician second), we come into this world with preconceived ideas, which are his ‘forms’. These ‘forms’ are an ideal perfect semblance from ‘heaven’, as opposed to the less perfect real objects in nature. This has led to the idea that anyone who prescribes to the notion that mathematical laws and relationships are discovered, must therefore believe in a Platonic realm where they already exist.
This aligns with the idea of God as mathematician. Herbet Westron Turnbull in his short tome, Great Mathematicians, rather poetically states it thus: ‘Mathematics transfigures the fortuitous concourse of atoms into the tracery of the finger of God.’ But mathematics does not have to be a religious connection for its laws to pre-exist. To me, they simply lie dormant awaiting an intelligence like ours to uncover them. The natural world already obeys them in ways that we are finding out, and no doubt, in ways that we are yet to comprehend.
Part of the whole philosophical mystery of our being and the whole extraordinary journey to our arrival on this planet at this time, is contained in this one idea. The universe, whether by accident or anthropic predestination, contains the ability to comprehend itself, and without mathematics that comprehension would be severely limited. Indeed, to return to my earliest point, which converges on Kant and Eco’s treatise in particular, Kant and the Platypus, our ability to comprehend the universe with any degree of certainty, is entirely dependent on our ability to uncover the secrets and details of mathematics. And consequently the limits of our knowledge of the natural world is largely dependent on the limits of our mathematical knowledge.
Addendum 1: This post has become popular, so I'm tempted to augment it, plus I've written a number of posts on the topic since. When studying physics, one is struck by the significance of scale in the emergence of nature's laws. In other words, scale determines what forces dominate and to what extent. This demonstrable fact, all by itself, signifies how mathematics is intrinsically bound into reality. Without a knowledge of mathematics (often at its most complex) we wouldn't know this, and without mathematics being bound into the Universe at a fundamental level, the significance of scale would not be a factor.
24 July 2025
The edge of time
This is a contentious idea, despite the fact that we all believe we experience it all the time. Many physicists, including ones I admire, and whom I readily admit know a lot more than me (like Sabine Hossenfelder), believe that ‘now’ is an illusion; or (in the case of Paul Davies) that it requires a neurological explanation rather than a physical one. I will go further and claim there is an edge of time for the entire universe.
I made the point in a previous post that if you go on YouTube, you’ll find discussions with physicists who all have their own pet theories that are at odds with virtually everyone else, and to be honest, I can’t fault them, and I’m pleased that they’re willing to share their views.
Well, I’m not a physicist, but this is my particular heretical viewpoint that virtually no one else agrees with, with the additional caveat that they all have more expertise than me. They will tell you that I’m stuck in 19th Century physics, but I believe I can defend myself against that simple rebut.
During COVID lockdown in 2021, I did a series of online courses through New Scientist, including one on The Cosmos, where one of the lecturers was Chris Impey (Distinguished Professor, Department of Astronomy, University of Arizona) who made the point that the Universe has an ‘edge in time’, but not an edge in space. He might have used the word ‘boundary’ instead of ‘edge’, which would be more appropriate for space. In fact, it’s possible that space is infinite while time is finite, which means that the concept of spacetime might have limited application, but I’m getting ahead of myself.
The one other person I’ve read who might (partly) agree with me is Richard Muller, who cowrote a paper with Shaun Maguire, titled Now, and the Flow of Time, as well as a book, NOW; The Physics of Time, which I’ve read more than once. Basically, the edge of time on a cosmic scale is the edge of the Big Bang (which is still happening). What I’m saying is that there is a universal ‘Now’ for the entire universe, which is one of the most heretical ideas you can hold. According to modern physics, ‘Now’ is completely subjective and dependent on the observer – there is no objective Now, which is what I challenge.
There is a way in which this is correct, in that different observers in different parts of the Universe see completely different things (if they’re far enough apart) and would even see different horizons for the Universe. In fact, it’s possible that an observer who is over the horizon to us will see objects we can’t see, and of course, wouldn’t see us at all. This is because objects over the horizon are travelling away from us faster than the speed of light.
Because the speed of light is finite, the objects that we ‘observe’ millions or billions of light years away, are commensurately that much older than we are. And it follows from this logic, that if anyone could observe Earth from these same objects, they would see it equally old compared to what we see. This means that everyone sees a different now. This leads to the logical question: how could an objective ‘now’ exist? I like to invoke Kant that we cannot know the ‘thing-in-itself’, only our perception of it.
And I invoke Kant when I look at relativity theory, because it’s inherently an observer-dependent theory. I would contend that all physics theories are epistemic, meaning they deal with knowledge, rather than ontic, which is what is really there. Some argue that even space and time are epistemic, not ontic, but I disagree. The dimensions of space and time determine to a large extent what sort of universe we can live in. A point made by John Barrow in his book, The Constants of Nature.
In a not-so-recent post, I explained the famous pole-in-the-barn paradox, where 2 different observers see different things (in fact, measure different things) yet, in both cases, there is no clash between the pole and the barn (or in the example I describe, a spaceship and a tunnel). One of my conclusions is that it’s only the time that changes for the 2 observers, and not the space. Instead, they measure a different ‘length’ or ‘distance travelled’ by using their clocks as rulers. But it also implies that one of the observers is more ‘privileged’ than the other, which seems to contradict the equivalence principle. But I can make this claim because there is a reference frame for the entire universe, which is provided by the CMBR (cosmic microwave background radiation). This is not contentious, because we can even measure our velocity relative to it by using the Doppler effect, hence our velocity relative to the entire universe.
But there is another famous and simple experiment that provides evidence that there is an overall frame of reference for the Universe, which philosopher of science, Tim Maudlin, called ‘the most important experiment in physics’. If you were to go to the International Space Station and spin an object, it would be subject to the same inertial forces as it would on Earth. So what’s it spinning in reference to? The spaceship, its orbit around Earth, or the entire cosmos? I’d say, the entire universe, which is obviously not spinning itself, otherwise it would have a centre. Of course, Einstein knew this, and his answer was there is no absolute time or space but absolute spacetime.
I raised this earlier, because, if time is finite and space infinite, the concept of absolute spacetime breaks down, at least conceptually. But space doesn’t have to be infinite to have no boundary. In fact, it’s either open and infinite or closed and finite, albeit in 3 dimensions. To provide a relatable analogy, the Earth’s surface is finite and closed, but in 2 dimensions. Marcus du Sautoy made the point that, if the Universe is spatially infinite, we might never know.
The other point is that you could have clocks running at different rates dependent on where they are in the Universe, yet there could still be a universal Now. This is implicit in the famous twin paradox thought experiment. I like to point out that when the twins reunite they have lived different durations of time, yet agree where they are in time together. This means you can have a universal Now for the universe while disagreeing on its age; if you lived near a massive black hole, for instance.
In the same way observers can travel different distances to arrive at the same destination, they can travel different time intervals as well. In fact, they would agree they’ve travelled the exact same spacetime, which is why relativity theory argues you can only talk about spacetime combined rather than space and time separately. But I argue that it’s the clock that changes and not space, where the clock is the ruler for space.
The fly-in-the-ointment is simultaneity. According to relativity theory, simultaneity is completely dependent on the observer, but again, I invoke Kant. There could be an objective simultaneity that can’t be observed. I’ve written on this before, so I’ll keep it brief, but basically, you can have a ‘true’ simultaneity, if both the observer and the events are in the same frame of reference. And you can tell if you’re not, by using the Doppler effect. Basically, the Doppler effect tells you if the source of the signals (that are apparently simultaneous) are in the same frame of reference as you. If they’re not, then they’re not simultaneous, which infers there is an objective simultaneity. Whether this applies to the entire universe is another matter.
You may be familiar with this diagram.
I want to make a couple of points that no one else does. Firstly, everything outside the past light cone is unobservable (by definition), which means relativity theory can’t be applied, yet people do. As I said earlier, relativity is epistemic and all epistemic theories (or models) have limitations. In other words, I contend that there is an ontology outside the light cones that relativity theory can’t tell us anything about (I discuss this in more detail in a post appositely titled, The impossible thought experiment).
Secondly, the so-called ‘hypersurface’ is a fiction, or at best, a metaphor. Yet Brian Greene, to give one example, discusses it and graphically represents it as if it’s physically real. If ‘Now’ is the edge of the Big Bang, it suffuses the entire universe (even if it’s physically infinite), which means it’s impossible to visualise.
Let’s talk about another epistemic theory, quantum mechanics. In fact, the ontology of QM has been an open debate for more than a century. I recently watched a discussion between Matt (from PBS Space Time) and Mithuna Yoganathan (of Looking Glass Universe), which is excellent. It turns out they’re both from Melbourne, which is where I’m writing this. I figured Mithuna was Aussie, even though she’s based in London, but I didn’t pick Matt’s accent. I have to admit he sounds more Australian in his conversation with her. Towards the end of the video, they readily admit they get very speculative (meaning philosophical) but Mithuna provides compelling arguments for the multiple worlds interpretation (MWI) of QM. Personally, I argue that MWI doesn’t address the probabilities which is intrinsic to QM. Why are some worlds more probabilistic than others? If all outcomes happen in some universe somewhere, then they all have a probability of ONE in that universe. If there are an infinite number of universes then probabilities are nonsensical.
If you go to 37.10m of the video where Mithuna talks about the Schrodinger equation and the ‘2 rules’, I think she gets to the nub of the problem, and at 38.10 puts it into plain English. Basically, she says that there are either 2 rules for the Universe or you need to reject the ‘measurement’ or ‘collapse’ of the wave function, which means accepting MWI (the wave function continues in another universe), which she implies without saying. She says the 2 rules makes ‘the Copenhagen interpretation untenable’. I find this interesting, because I concluded many years ago that the Universe obeys 2 sets of rules.
My argument is that one set of rules, determined epistemically by the Schrodinger equation, describes the future and the other set of rules, which is classical physics and is determined by what we observe, describes the past.
A feature of QM, which separates it from classical physics, is entanglement and non-locality. Non-locality means it doesn’t strictly obey relativity theory, yet they remain compatible (because you can’t use entanglement to transmit information faster-than-light). In fact, Schrodinger himself said that “entanglement is the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.” In other words, it obeys different rules to classical physics, with or without ‘measurement’.
MWI effectively argues that superposition exists in reality, albeit in parallel universes, whereas I contend that it only exists in the future. The wave function describes all of these possibilities, and via the Born rule, gives them probabilities. But when we observe it, which axiomatically puts it in the past, there is only ONE and there is no longer any superposition.
All physicists agree that entanglement, in principle, can apply to objects on opposite sides of the Universe. In fact, Schrodinger’s equation, in principle, can describe a wave function for the entire universe, which is why I’ve half-jokingly called it God’s equation, and have it tattooed on my arm.
I contend (though, as far as I know, no one agrees with me) that entanglement across the entire universe only makes sense if there is a universal Now for the entire universe. A Now that separates QM future superpositions (described by the wave function in Schrodinger’s equation) from past ‘observables’ in classical physics.
22 January 2008
Is mathematics evidence of a transcendental realm?
I am currently reading a book called Thinking about Godel and Turing by Gregory J. Chaitin. This book, I must admit, is slightly over my head, so I walk on tippy toes holding my head back in order to keep my nose above water. I read a lot of books by people who are much cleverer than me, but then I guess I am the audience they are writing for. Richard Feynman was once called the smartest person in the world, or some such honorific, by OMNI magazine (the first issue if I recall correctly), which suitably embarrassed him, but he was one of the great physicists of his generation, if not the greatest, and yet also one of the greatest teachers. I have read no one who writes so well for people with lesser abilities than himself. I can think of other writers: Roger Penrose comes to mind, who can write for people less clever than himself; Paul Davies is another, and one would also have to include Stephen Hawking. These are to be contrasted with other academic writers I have read, who do their best to show how much cleverer they are than their readers, but risk misleading them by talking authoratively on topics outside their field. I am not an academic and I have no expertise, so, if I be so charged, I stand guilty. I am the first to admit that I am not as clever as I may appear. I am intellectually curious and I can write well, that is all. And yes, I am provocative – no apologies there.
Gregory Chaitin’s book is really a collection of essays, often transcripts of lectures or public addresses he’s given over the past 30 years. As such, he is also writing for people less clever than himself, and I think he does a commendable job. He also generously acknowledges his heroes (especially Leibniz). I only hope I don’t misinterpret him in my attempt to glean something philosophically meaningful from his text. I have to say that I found his ideas, and his exposition of them, exciting to read.
The 20th Century will be remembered for a number of things: Peter Watson performs an excellent job documenting many of its achievements in art and science in a narrative form with his magnum opus, A Terrible Beauty. (Another writer who knows how to illuminate without his ego intruding in the process.) I’ve had disagreements with Watson, philosophically, and we’ve had brief correspondence, but I think this book is an achievement of almost heroic proportions, not least because he can write with equal erudition on art and science.
For my mind, the 2 outstanding events of the 20th Century, which will be remembered throughout human history, are powered flight leading to exploration beyond our planet and the invention of the computer with all its consequences. But Chaitin rightly points out that there were 2 revolutions that occurred early last century, at about the same time that humankind took flight in a literal sense, that will also be remembered as historical milestones. I’m talking about Einstein’s theories of relativity and the development of quantum mechanics. In addition to these revolutions, Chaitin adds a third: Kurt Godel’s proof of his ‘Incompleteness Theorem’ in mathematics (1931) and Alan Turing’s related theorem concerning the so-called ‘halting problem’ for computers (1936). This particular revolution requires some elaboration.
Firstly, when Turing developed his thesis, computers didn’t exist, and, in fact, Turing’s paper is better remembered for containing the purely conceptual idea of the ‘Universal Turing Machine’, which is what all modern computers are, including the one I’m now writing on. So this revolution is directly related to the more concrete revolution I referred to in the opening of my last paragraph. But Chaitin’s point, that this revolution, first enumerated by Godel, being of equal significance as relativity theory and quantum mechanics, should not be lost.
But before I continue on this theme, I would like to say something about Turing, one of my heroes. Turing is probably best known for his pivotal role in breaking the ‘enigma’ code during WWII, but I would also hope he be remembered for his tragic death, so that ignorance and prejudice would not claim such a brilliant mind in the future. Turing was one of the greatest minds of the 20th Century, arguably, second only to Einstein. Turing may not have physically invented the computer (a moot point), and it certainly would have evolved without him, in the same way that Einstein didn’t invent the Lorentz transformations that lie at the heart of relativity, and certainly relativity’s consequences would have also been discovered without him. But both men thought outside the square in a way that goes beyond cliche, and both men were ahead of their time by at least a generation, and both men were undisputed geniuses.
Turing’s death, however, is comparable to the deaths of two other great minds of science and philosophy: Socrates and Lavoisier. Socrates (arguably the father of Western philosophy) was forced to suicide for political reasons, and Lavoisier (the ‘father of chemistry’) was guillotined in the aftermath of the French Revolution. All these deaths were the result of political and social forces present at the time, and all were regretted almost immediately afterwards. One might say that they were all victims of ignorance, and that includes Turing. Turing suicided by eating an apple injected with cyanide after he was legally prosecuted for being homosexual (he was blackmailed first) and forced to take hormonal treatment that had him growing breasts. This should be kept in mind when we have conservatives in both religion and politics who think the legal attitude towards homosexuality has been ‘socially disastrous’ (Cardinal George Pell quoted by a reviewer of his latest book, God and Caesar). Unfortunately and tragically, Turing was born ahead of his time in more ways than one.
A philosophical detour onto another path, we’re now back to the topic at hand. Not quite 10 years ago, when I was studying philosophy in an undergraduate course, I had to write an essay on Immanuel Kant with the subject: What is transcendental idealism? In preparation I read large parts of his seminal work, The Critique of Pure Reason, about as dense a text as one could find. Apparently Kant’s lectures were very popular and much more accessible than his writings. Unfortunately, he lived before the age of electronics, otherwise we might have transcripts of his lectures rather than his essays. The Critique of Pure Reason is the only book I’ve read that contains at least one sentence over a page long. You may be wondering what this has to do with Godel and Turing: well, Kant wrote a great deal about epistemology which is effectively the subject of Chaitin’s book. Of more relevance to this discussion, is my conclusion in that essay: if there is a ‘transcendental idealism’, it must be the world of mathematics.
I won’t reiterate my arguments here (perhaps a future posting), but it suggests a starting point for the question that heads this essay. Kant understood that our knowledge, our perception and our interpretation of the world had 2 components: an empirical component based on experience and an ‘a priori’ component based on reasoning and imagination. It is this latter component that leads to the concept of ‘transcendental idealism’ and has more than a passing resemblance to Plato’s ‘forms’. I don’t wish to get too esoteric about this, so I will present the same idea in a more prosaic context. I’ve said elsewhere that the success of science is a direct consequence of a continuing dialectic between theory and experiment, or theory and observation. This is exactly the same thing that I believe Kant was talking about, keeping in mind that he lived in the time following Newton when it was believed we all existed in a clockwork universe.
My own particular take on this is that mathematics is the principal medium that allows this dialectic to occur. Without mathematics our comprehension of the universe (the entire natural world in fact) would be limited in the extreme. This realisation, in Western philosophy at least, began with Pythagoras, was given impetus by Galileo, Kepler, Newton and Leibniz (along with many others), but only found it’s true significance in the 20th century, with Einstein (following Maxwell), along with Bohr, Schrodinger, Heisenberg and all those who have contributed since. Chaitin’s book, as I’ve already said, is effectively about epistemology and, in particular, the epistemology of science and mathematics. In fact, Chaitin’s entire thesis is that they are more closely related than we tend to think, but I’m getting ahead of myself.
Firstly, I need to share with you Chaitin’s excitement, and sense of historical significance, that he finds in Godel’s Incompleteness Theorem of 1931. Before this theorem, and even after it, mathematicians have believed that mathematics is inherently axiomatic (going back to Euclid, another Greek), which is its strength and its claim to objective truth. But even before Godel, as Chaitin points out, Georg Cantor and Bertrand Russell had already shown that mathematical certainty could be a chimera. Cantor is best known for his ‘diagonal method’ of showing why there are more ‘real numbers’ than ‘rationals’ (Penrose gives a good exposition in The Emperor’s New Mind). Turing, by the way, employed Cantor’s diagonal method in the most critical step of his ‘halting problem’ proof, so it has far-reaching consequences.
Over a hundred years ago, Cantor postulated the idea of infinite sets (transfinite numbers), which was such a radical and controversial idea for its time, that, according to Chaitin, Cantor suffered a breakdown as a result of the criticism and was never given a position in a first rate institution. Being ahead of your time can sometimes be a career stopper, no matter what your achievements. These days, Cantor is regularly referred to in mathematical texts on number theory.
Godel gave a proof, that took the whole mathematical world by surprise, that the so-called axiomatic method was flawed, or, at the very least, could not be unconditionally relied upon. Effectively, Godel’s proof and Turing’s, which is even more demonstrative, says that, no matter what formal mathematical system you have, based on a set of known axioms, there is always the possibility of mathematical ‘truths’ that cannot be derived from these axioms. So the method of determining mathematics that we have all relied upon since the concept of numbers was derived, is not so deterministic after all. Now, as Chaitin points out, despite the absolute shock this conclusion created, people have largely carried on as if it never happened. Many people see it as an esoteric anomaly that has no bearing on real mathematical problems, but, as Chaitin points out, that is not the case.
The best example would be Reimann’s hypothesis and the Zeta function. There have been some excellent books written on this subject (Prime Obsession by John Derbyshire, The Music of the Primes by Marcus du Sautoy and Stalking the Reimann Hypothesis by Dan Rockmore are three I enjoyed reading). I won’t elaborate, except to say that it is a convoluted and intriguing journey into the mathematical realm, and it is to do with the distribution of primes, but it’s the perfect example. It’s the perfect example because computer programmes (Turing machines) have calculated it to be correct to astronomical magnitudes, but there is still no proof. It demonstrates perfectly the so-called ‘halting problem’ because if the programme halts the hypothesis is false, and if the hypothesis is correct, then the programme will never stop (unless instructed to of course). But more than this, most mathematicians accept it as true, despite the lack of a ‘formal’ proof, and it is now used as an ‘axiom’ for other mathematical proofs, albeit conditionally. And this is what Godel said, that there can be an axiom, or axioms, outside the formal system you are using that can be the basis of newly discovered mathematical ‘truths’. Another, more readily comprehended example, also given in Chaitin’s book, is Goldbach’s conjecture: all even numbers above 2 are the sum of 2 primes. (You can check this for yourself with the first 10 even numbers, remembering that 1 is not considered a prime.) A relatively simple computer programme can be written to check this, but, again, it only stops if the conjecture is wrong. (This has been checked to 10 raised to the power of 14, 1 with 14 zeros after it).
Now, strictly speaking, what I have just described isn’t the halting problem, but a consequence of it. What Turing said (proved, in fact) is that there is no way of knowing if a programme will halt or not for a particular theorem. If we knew that, then, obviously, we would be able to say in advance if these conjectures were true or false.
Chaitin makes the comparison between this discontinuity of axioms and physics. He gives the example of Maxwell’s equations having no basis in Newton’s equations, yet forming an ‘axiom’ for Einstein’s equations of relativity. Likewise, quantum mechanics has no basis in either Newtonian mechanics or Einstein’s relativity, but has become a new ‘axiom’ for future theories (Thomas Kuhn calls them paradigms). Chaitin argues, that not only does this demonstrate that mathematics and physics are more closely related than we consider, but that there is good reason to suggest that mathematics should be done more like physics, where new axioms may not have to rely on previous ones. Chaitin calls this proposed methodology ‘quasi-empiricism’, a term coined by Imre Lakatos.
Chaitin goes even further on this subject, and claims that the similarity between physics and mathematics lies at their base, which is randomness. In fact, Chaitin claims that this is his major contribution to mathematics, arising from his invention of the term ‘Ω’ (Omega), though he calls it a discovery, to designate the probability of a programme ‘halting’, otherwise known as the ‘halting probability’. I won’t elaborate too much on this, so, if you want to know more, you will need to read his book. For Chaitin, ‘Ω’ is the logical extension of Godel’s and Turing’s landmark theories, and proof of mathematics’ inherent irreducibility (his term). The significance of this ‘discovery’, according to Chaitin, is that it’s proof that there is no mathematical ‘theory of everything’ (TOE) – no all encompassing meta-mathematical theory. But he sees this as liberating. To quote: ‘Ω shows that one cannot do mathematics mechanically and that intuition and creativity are essential.’
Another person who discusses these issues (raised by Godel and Turing) in devoted detail, is Roger Penrose (The Emperor’s New Mind), but in the context of Platonism. Penrose is a self-confessed ‘Platonist’, meaning he believes that mathematics exists in an independent realm to the human mind. This is a contentious viewpoint (I discuss it from a different perspective in my Sep.07 posting: Is mathematics invented or discovered?). Chaitin says very little on this question (see below), but quotes Godel, who was a ‘Platonist’, and Einstein, who was not. Paul Davies, who writes an excellent foreword to Chaitin’s book, makes the case, in a couple of books, (The Mind of God and The Goldilocks Enigma) that mathematics ‘shadows’ the natural world, but doesn’t call himself a Platonist. Stephen Hawking, who famously worked with Penrose on singularities and black holes, doesn't share his colleague's philosophical viewpoint at all, and calls himself an 'unashamed reductionist' and a 'positivist'. Most philosophers dismiss the notion of Plato’s forms, but mathematics is an area where it persists. I dislike the term but I agree with the philosophical premise: mathematics has an independent existence to human thought. Plato’s forms originally applied to everything, not just mathematics, so somewhere there was a perfect world (like heaven) and Earth was merely a facsimile of it. This is similar to some people’s interpretation of Taoism, but it’s not mine. But this brings me to the subject alluded to in the title of this posting: mathematics is arguably the only evidence we have of a transcendental, or metaphysical, realm.
Interestingly, people on both sides of this argument present Godel’s famous Incompleteness Theorem as supporting their philosophical point of view. Chaitin himself says, ‘[Godel’s theorem] exploded the normal Platonic view of what math is all about’, without elaborating on what he means by ‘normal Platonic view’ in this context. Russell, according to one account I read, was derisively disappointed when he met Godel and discovered he was unashamedly a Platonist. Many people I’ve met, philosophers in particular, believe that Russell and Wittgenstein settled this question for good, but I’m not sure that many physicists would agree. I once had a conversation with a philosophy lecturer, whom I greatly respected, who asked me if I thought that mathematics done by some hypothetical inhabitants in the constellation Andromeda would be the same as mathematics done by us on Earth. I answered: Of course; to which he responded: But you’re assuming that Andromedans would use base 10 arithmetic. I said that this is like saying that a tiger in China is not a tiger because it is called something else in Chinese. I used this analogy because he had used it himself to make an epistemological point earlier in the discussion. Unfortunately, he just assumed that I didn’t know what I was talking about, and I never got the opportunity to enlighten him further.
Using the same hypothetical, Chaitin quotes Stephen Wolfram (A New Kind of Science), whom I haven’t read, who argues, and gives examples, of mathematics that might be different to what we are familiar with. But I would suggest, that unless the laws of the universe are significantly different on another planet, then the mathematics any inhabitants developed would be the same as ours. Because, as Davies and Penrose point out, mathematics and the natural world are married in a way that is inescapable to anyone who explores them deeply enough. Even on our own planet, different cultures developed mathematical ideas independently but were ultimately convergent. So whilst I agree that mathematics may be a boundless realm, its marriage to the natural world suggests inevitable avenues of investigation and discovery.
Penrose, in particular, argues a very strong case for Platonism. In The Emperor’s New Mind, he spends an entire chapter on the Mandelbrot set (with a detour to Cantor, Euler and Gauss) and presents it as an exemplar of Platonist mathematics. The entire Mandelbrot set exists only in an infinite realm so that no one will ever see it in its entirety, yet it is generated by a simple algorithm or formula. (This leads to a discussion on complexity, which is also a key theme in Chaitin’s book, but I will return to complexity in a moment.) For Penrose, the Mandelbrot set is evidence that something can only exist in a mathematical realm that we only get a glimpse of – this is a very profound idea. (To get a glimpse, check the following link: Mandelbrot Set ) Is this different to any other work of art? Well, Penrose makes the same analogy, but the fundamental difference is that mathematics doesn’t manifest itself as a ‘unique’ or ‘one-off’ creation, as works of art do. (Someone else could have discovered Reimann’s geometry or Schrodinger’s equations, but no one else could have created Beethoven’s symphonies or Bach’s Brandenburg concertos). And it is difficult to escape the connection between mathematics and the natural world, the Mandelbrot set notwithstanding.
In any discussion on mathematics, including Chaitin's, one cannot escape infinity – it infiltrates all attempts to capture it and tie it down. It’s also what makes it elusive (take the Reimann hypothesis) and boundless in every sense (look at Î and e). It’s what takes it outside human experience and makes it ‘magical’ (like the calculus). In Euler’s famous equation, infinities abound, yet it’s a simple relationship between e, Î , i, 1 and 0 (where i is the square route of minus 1). Feynman called it ‘the most remarkable formula in math’ when he thought he had ‘discovered’ it a month before his 15th birthday. ( See link: Euler's Equation ) To appreciate the complexity that lies behind this simple equation, and the way it ties together so many branches of mathematics, you need to go to Euler's Formula.
For Penrose, it’s almost religious:
‘The notion of mathematical truth goes beyond the whole concept of formalism [this is Godel’s theorem in a nutshell]. There is something absolute and “God-given” about mathematical truth. This is what mathematical Platonism… is. Any particular formal system has a provisional and “man-made” quality about it… Real mathematical truth goes beyond mere man-made constructions.’
Strong words indeed. This has been a lengthy treatise, but not one that is especially decisive or well-argued. I have hardly touched the subject of complexity, which is a key component of Chaitin’s thesis, indeed his life’s work. One of the points he makes is that mathematical complexity may provide a key to understanding biological evolution – after all, DNA is the world’s most extraordinary piece of software. Complexity, as described by Chaitin, is effectively the difference in the length of an algorithm (in bits) to the length of the results it produces (he defines it in a logarithmic expression). The Mandelbrot set is a good example, because a very short algorithm can produce an extraordinarily detailed and complex picture of infinite proportions via a computer. DNA is, in effect, a very small molecular structure that can produce extremely complex and diverse organic entities that have life (ad infinitum it would appear); so I would argue that it’s more than just an analogy. (Chaitin makes the point that, with its 4 bases, human DNA contains 6 trillion bits of information; 6 followed by 9 zeros.)
Perhaps there is another level of complexity behind DNA in the same way that quantum mechanics exists behind classical physics. No one can anticipate what we will find. When Darwin hypothesised about evolution, no one would have predicted genes, let alone DNA. And when Newton proposed gravity no one would have predicted relativity theory, let alone quantum mechanics. We think, just like they did, that we’ve discovered everything there is to discover, but we haven’t.
This essay only scratches the surface of Chaitin’s multi-layered thesis, so, if it stimulates you, read his book. My favourite chapter is titled On the intelligibility of the universe, where he liberally quotes great minds like: Einstein, Feynman and Born; all ruminating on the theoretical component of the dialectic of science that I referred to earlier.
Does Chaitin believe in a mathematical transcendental realm? Well, he certainly believes in a metaphysical approach, subscribing to a “digital philosophy” (his quotation marks), along with Edward Fredkin and Stephen Wolfram. He calls it a ‘neo-Pythagorean vision of the world’, where ‘God is not a mathematician’, but ‘a computer programmer.’ But he adds the following caveat: this is a new viewpoint, and it will be interesting to see how far it takes us.
Personally, whilst I don’t have the intellectual abilities of these people, and therefore I can’t challenge their premise, I believe there is more to the universe than algorithms. For a start, I don't believe the human mind runs on algorithms, despite what some cognitive psychologists might think (on that point I agree with John Searle and Roger Penrose).
So Chaitin argues that most real numbers are uncomputable and this makes mathematics infinitely complex (if a number can't be calculated there is no formula or algorithm for it, which makes it infinitely complex by Chaitin's own mathematical definition of complexity, though he credits Leibniz with the original idea). Also, I accept his argument that there is no overall meta-mathematical theorem - no TOE for mathematics - because that is the essence of Godel's and Turing's proofs. I agree with his statement that intuition and creativity are essential, because history has demonstrated that beyond dispute. I would not be surprised if, as he speculates, mathematics gives us an unexpected insight into biology and evolution, though, obviously, I've no idea how it might happen. And, as I have said elsewhere, I believe it is our knowledge of mathematics that will determine the limits of our knowledge of the physical universe and the natural world. In my opinion, this was Pythagoras's great paradigmatic insight and his legacy to philosophy and science.
Mathematics can take us into worlds that we don’t normally perceive: higher dimensions, complex planes, infinite series and infinitesimal intervals – but in the world we live in, it continues to uncover riches and mysteries beyond our imagination.
You may also want to read my post on The Laws of Nature (Mar.08).