The Case Against Reality by Donald D. Hoffman

As I’ve pointed out before, YouTube has allowed scientists and philosophers of all stripes to promote pet theories that they know they can’t pursue academically with the same freedom. In fact, I’ve heard comments from people like Gregory Chaitin and Sabine Hossenfelder, who feel that the innate conservatism in academia is hindering progress in research and exploring new ideas.

 

Donald Hoffman is possibly an exception given his credentials: ‘Professor of Cognitive Sciences at the University of California, Irvine… [He] is the author of over one hundred scholarly articles on various aspects of human perception and cognition. He received a Distinguished Scientific Award of the American Psychological Association for early career research into visual perception, the Rustum Roy Award of the Chopra Foundation and the Troland Research Award of the US National Academy of Sciences.’

 

All of the above I’ve lifted verbatim from the About The Author section to his book, The Case Against Reality (2019). The copy I have is Penguin paperback (2020). The subtitle is How evolution hid the truth from our eyes. I first came across Hoffman 10 years ago (2016), when someone sent me a link to an academic paper he cowrote with Chetan Prakash called Objects of Consciousness, which I reviewed back then.

 

All of this gives Hoffman enormous credibility, without even mentioning that he had direct communication with Francis Crick before the latter died. His book is obviously aimed at people like myself, who have an interest in science but are not academic, with the intention of convincing us that his ideas are unassailable. But the truth is that by the time I reached the end of his book, I was more convinced that he was wrong than when I started. Now that’s a huge admission given the extraordinary gap in our credentials not to mention the limitations of my knowledge in neuroscience.

 

I wrote a post some time ago, where I talked about how important beliefs are in the development of science. What I’ve learned from watching various science experts on YouTube, who have, what one might call, ‘fringe ideas’, is that they form a very strong belief and then look for the evidence to support it. I include myself in this, even though I’m not a scientist. My fringe idea is that there is a universal now, and I think this is consistent with relativity theory, even though everyone will tell you that I’m wrong. The one thing going in my favour is that the Universe appears to have an edge in time and a finite age, which runs contrary to the orthodox view that there is no objective now. From what I’ve seen and read, physicists gloss over this conundrum, including Brian Greene in his excellent book, The Fabric of the Cosmos.

 

A better example is cosmologist Claudia de Rahm, who is one of the very few scientists (perhaps the only one with the necessary expertise) who thinks that gravitons could have mass. The point is that we don’t even know if gravitons exist, and she readily acknowledges, that if they do exist, they might be impossible to detect.

 

So I can’t criticise Hoffman simply because his ideas are ‘out there’; it’s not uncommon among scientists, including famous ones like Sir Roger Penrose.

 

This is a lengthy preamble, but I want to make one other point, before I get into the nitty gritty of his arguments. I wrote a post once on how I believe science works, which, in a nutshell, builds on what we already know, even when we have so-called revolutions in science. Ever since the scientific revolutions of Copernicus, Galileo, Kepler, and of course, Newton and Leibniz, our theories have extended science rather than outright replaced what went before. Claudia de Rahm (cited above) in answer to an audience question I watched online, made the explicit point that whatever new discoveries we make, they have to explain what we already know, so we don’t just throw everything in the rubbish bin (to use her turn-of-phrase).

 

Many people don’t know that one of Einstein’s self-imposed constraints on his General Theory of Relativity (GR) was that it had to agree mathematically with Newton’s theory of gravity when relativistic effects were negligible. Likewise, quantum mechanics (QM) hasn’t so much replaced classical physics, as added to it and possibly underpins it. UK based Aussie physicist, Mithuna Yoganathan (from the Looking Glass Universe YouTube channel) has made the point that she’s attracted to the many worlds interpretation (MWI) of QM because she can’t accept that the Universe obeys 2 sets of rules: for quantum mechanics and classical physics. The fact that she even mentions it suggests to me that it’s quite plausible, in which case, QM adds new physics without completely overturning the old.

 

The reason I’ve spent so much time on this detour is that, according to Hoffman, we need to reinvent virtually all the physics of the past 4-500 years, as well as evolutionary theory.

 

In fact, Hoffman states that “ITP asserts that one theory of objective reality – that it consists of physical objects in spacetime – is false.” (p.82) Einstein’s GR, mentioned earlier, is one such theory and currently underpins the latest cosmological model, specifically ΛCDM. ITP is Hoffman’s ‘Interface Theory of Perception’ which he analogises with a desktop interface on a computer, and to extend the analogy, the ‘objects’ we perceive are like ‘icons on the desktop’, so not ‘real’. I’ll return to this point later, given its significance to his argument.

 

After reading his book, I have many philosophical differences with Hoffman, but there is one point in particular, which is central to his entire thesis, where I’d argue the evidence contradicts him. I’m calling it the ‘principle of object permanence’. I admit I made that up, but it’s something that I argue evolution by natural selection selected for, because it’s a principle we not only take for granted but is cognitively found in babies and other animals. In other words, it’s the fundamental belief that an object still exists when we can’t see it. I will give an example of its ‘evolutionary value’ shortly, but it’s the direct opposite of what Hoffman believes. And I mention its evolutionary value because key to Hoffman’s theory is that evolution by natural selection supports his position and not mine.

 

He calls his theorem FBT, ‘Fitness-Beats-Truth’, which effectively means that evolution selects for fitness, not truth, while being somewhat hand-wavy about what those terms mean and how they can be quantified. The one example he provides in his book is a graphic representation. In fact, all his ‘evidence’ is based on computer modelling according to the examples he gives. He combines FBT with ITP (already mentioned), asserting they prove that he’s right: object permanence does not exist (as I defined it above).

 

I’m not going to try and prove his FBT or ITP wrong; I’m going to show why I believe the principle of object permanence is a necessary belief for evolutionary fitness. Now Hoffman would agree that the perception of object permanence affects evolutionary fitness but that doesn’t mean it’s real. However, I contend that it can make the difference between life and death, which does make it real.

 

I will give an example I read about once, involving crows not humans, which I assume really happened. Actually, it involved crows and humans, but it’s the cognitive ability of the crow that’s tested. A human with a gun set a trap for a crow using bait, then went into hiding until the crow took the bait. The crow, however saw the man go into hiding, and believing in object permanence waited patiently till the man left. Then the man got a companion, and this time the crow waited until both of them left. So they repeated the exercise using 3 gunmen and this time after 2 left, the crow thought it was safe, only it wasn’t. The exercise proved that the crow couldn’t count past 2. But you can see how a belief in object permanence can be critical to survival.

 

Now Hoffman has an answer to this, because it’s comparable to being hit by a car or being killed by any so-called object in his ‘interface’ (hence the name, ITP). He would say you need to take the car ‘seriously but not literally’. I admit I have a problem with that statement: I take the car ‘seriously’ because it can kill me but I don’t take it ‘literally’ because it’s not ‘real’. Hoffman keeps comparing this to someone playing Grand Theft Auto on a computer, because we live in an ‘interface’ not spacetime and there are no objects, only ‘icons’, including your own body. He later argues we don’t live in a computer simulation, which I’ll return to.

 

For examples of truth that evolution didn’t select for, he mentions oxygen and ultraviolet light causing sunburn. His point is that in both cases we don’t need to know about oxygen in order to know that the ability to breathe air confers fitness or know about ultraviolet light to know that keeping our skin covered against prolonged exposure to sunlight also confers fitness. By these examples, truth means explanations, to which I’d agree: evolution doesn’t select for explanations and I wrote about that elsewhere, while also discussing Hoffman. Note how sunlight and air exist externally to us.

 

Returning to my principle of object permanence, Hoffman cites Einstein’s famous retort to Bohr, ‘Do you think the moon ceases to exist when you’re not watching?’ It turns out that Hoffman does: “There is no objective moon or spacetime that exists even when unperceived…” (p.198)

 

I would like to know how that affects the tides occurring all around the world, although, according to Hoffman, there is no ‘around the world’, because there is no 3-dimensional space, and there is no cause and effect. He also claims that DNA and chromosomes don’t exist unperceived which turns evolutionary theory on its head. I discussed that in my original post on Hoffman. It reinforces my earlier point that Hoffman’s theory requires all of science to be replaced, including evolutionary theory as well as cosmology. He doesn’t explain how DNA and chromosomes can only exist when observed, yet underpin evolutionary theory as we currently know and understand it. He also claims the same for neurons in the brain, meaning consciousness creates them rather than the converse, which is the consensus of virtually all neuroscientists. Hoffman acknowledges this fundamental disagreement.

 

He spends a lot of time discussing optical illusions in considerable detail as supporting evidence that what we perceive is not real. I agree with him that our sensory experiences, like colour, sound and smell are all totally subjective and that other animals’ sensory experiences can be completely different to ours. We know that some animals can see colours we can’t and vice versa, and some can hear sounds that we can’t. But here’s the thing: colours are created by reflected light from an external source, and sounds are caused by vibrations transmitted through a medium, like air, and we’re evolutionarily adapted to pick them up and transform them into sensory experiences. To use his terms, the ‘fitness’ is our sensory experience and the ‘truth’ is our explanation of their external origins. So, yes, we evolved for fitness, not a scientific understanding called truth, but that doesn’t rule out object permanence (as described above) nor their location in space and time.

 

For the sake of brevity, I can’t address all his points but there is one other I find particularly contentious, and that is the idea that 3 dimensions of space are redundant, and we only need 2. Hoffman cites the cosmological holographic principle, which is basically the idea that all the (quantum) information inside a black hole is available holographically on the surface of its event horizon, which, it needs to be pointed out, is a sphere. Some argue that our entire universe could be modelled on the same principle, that everything in it could be a hologram projected from a 2-dimensional horizon. Hoffman argues that since all the information can be contained in 2 dimensions the third dimension of space that we ‘perceive’ is redundant and is used for ‘error-correction’. Aside from this being highly speculative physics, Hoffman overlooks the fact that the 2-dimensional surface of an event horizon on a sphere can only exist in 3-dimensional space; a hollow sphere in 2 dimensions becomes a circle.

 

But there is more: the inverse square law for gravity, which is in both Newton’s and Einstein’s theories, is a direct consequence of space being 3-dimensional. John Barrow gives the best account I’ve come across in his excellent book, The Constants of Nature. Given Hoffman’s access to academia, I’d recommend he talk to Barrow, or at least, read his book. Hoffman quotes Arkani-Hamed in a 2014 lecture at the Perimeter Institute: “Almost all of us believe that spacetime doesn’t exist; that spacetime is doomed, and has to be replaced by some more primitive building blocks.” (p.114) Now, I’m obviously not familiar with Arkani-Hamed’s work, but Graham Farmelo (The Universe Speaks in Numbers) also quotes Arkani-Hamed in a different context, which is the success of mathematical objects called amplituhedrons in predicting the amplitude of gluons in particle physics. Note this has an experimental physics basis, and is not just theoretical.

 

This is a concrete example of a way in which the physics we normally associate with space-time and quantum mechanics arises from something more basic. (my emphasis)

 

Now, I know that some physicists are speculating that spacetime might be an emergent property from something deeper (as inferred in the above quote from Arkani-Hamed), but that’s not the same as saying it doesn’t exist. If, on the other hand, Arkani-Hamed really believes that ‘spacetime doesn’t exist’, then he needs to replace the highly successful GR with something else. He also needs to explain the cosmological history based on the observable CMBR (cosmic microwave background radiation) that is the remnant of the Big Bang (13.8 billion years ago). I don’t know how you can have a cosmological history without spacetime. Hoffman challenges this by arguing for a ‘top-down cosmology’, that starts with an observer, but that’s another topic for another time.

 

In fact, Hoffman argues that conscious agents generate spacetime and all the physics we currently know, including quantum mechanics.

 

Conscious realism must ground a theory of quantum gravity, explain the emergence of our spacetime interface, and its objects, explain the appearance of Darwinian evolution within that interface, and explain the evolutionary emergence of human psychology. (pp.198-9)

 

We must show how conscious agents generate spacetime, objects, physical dynamics and evolutionary dynamics. We must get back quantum theory and general relativity, and generalisations of these theories that are mathematically precise. (p.184-5)

 

Hoffman does this by creating a mathematical model of consciousness that apparently generates all of what we call objective reality. He reveals this in an appendix, titled, Precisely; The Right to Be Wrong. He doesn’t make it clear who has ‘the right to be wrong’: himself or the rest of us. It’s very obtuse mathematics, in the form of logic rather than equations, but basically, he creates a mathematical formula for ‘conscious agents’, and then, using ‘Markovian dynamics’, effectively derives spacetime. I don’t pretend I can follow it. He concludes by saying: “…perhaps a dynamic evolution of conscious agents toward small-world networks may appear in spacetime as the dynamics of gravity.” (p.205) I think the words, ‘perhaps’ and ‘may’, carry a lot of weight. To provide context, Hoffman argues that it’s networks of ‘conscious agents’, not individuals, that generate the world we call reality.

 

Earlier this year, I attended a 2hr stage presentation by Prof Brian Cox, which was very esoteric, discussing wormholes, black holes and other cosmological exotica, but he never mentioned that ‘spacetime is doomed’; so that’s the orthodox view.

 

In summary, I have 2 fundamental contentions with Hoffman, without even considering his ideas on evolution. Anyone who has lost an object (like car keys) knows they still exist unobserved, including Hoffman, I suspect. He will rationalise that he takes them ’seriously but not literally’. However, that explains nothing, unless you’re in a video game (refer below). My argument is that evolution confers fitness on perceiving ‘object permanence’ because it’s a matter of life-and-death, as per my crow story.

 

My second contention is that we live in 3D not 2D. To demonstrate, draw an animal, any animal, on paper, so it’s flat and you can cut it out. Then draw an alimentary canal from its mouth to its anus. If you then cut it out, it falls apart into 2 pieces. This is as good a reason as any why we don’t live in a 2D universe. Hoffman mentions, in passing, while discussing something else, that he can’t visualise 4D, and there’s a good reason for that: we don’t need to; even though we can represent any finite dimensional world mathematically. The reason we can only visualise 3D and less, is because we live in a 3D world.

 

Hoffman finishes his main text (before the Appendix) with this:

 

Spacetime is your virtual reality, a headset of your own making. The objects you see are your inventions. You create them with a glance and destroy them with a blink. You have worn this headset all your life. What happens when you take it off? (p.202)

 

Despite claiming that he doesn’t believe we live in a computer simulation, his description of ‘reality’ is indistinguishable from one. This is demonstrated by calling objects and even his own body, ‘icons’, and that he believes the world is 2D like a screen, which we erroneously perceive as 3D (according to him), because it increases our evolutionary fitness.

 

He reinforces that view here: “The interface theory of perception contends that there is a screen – an interface – between us and objective reality.” (p.191, my emphasis)

 

Later he talks about spirituality, which I won’t elaborate on, except to say that the idea that consciousness can escape its physical constraints, as opposed to creating them, is not ruled out or ruled in.

An excursion into the absurd with Camus as a guide

Someone lent me a book on Camus, titled Albert Camus and the literature of revolt. It was written by John Cruikshank, ‘the first professor of French at the University of Sussex; founding the French studies department at that institution in 1962’ (Wikipedia). It was first published in 1959, but I believe the edition I had was the ‘third Galaxy printing, 1963’, even though the copyright says 1960 just to confuse everyone.

 

Camus died 4 Jan 1960, but the author references him consistently as still alive, which is what you’d expect if it was written in 1959. It’s a very dense read, even for an academic. I’m not exaggerating when I say I found Kant’s Critique of Pure Reason an easier read (mind you, that was over 25 years ago).

 

I read 2 of Camus’ novels in high school (in the 1960s): The Plague (in English) and L’Etranger (in French). The Plague had quite an effect on my fertile brain at the age of 16, because it challenged a short lifetime’s indoctrination in Christianity. To be fair, I was already challenging the concept of God I had been raised on, so I think my mind was ripe. I’ve read it twice since, which says a lot. I saw the recent movie, The Stranger, based on the novel, L’Etranger, and I have to say I thought it was very well done, even down to the use of black-and-white visuals to capture the period, and found it very true to my memory of it.

 

Camus is famous for his philosophical position on the absurd, which is effectively the subject of Cruickshank’s tome over 224 densely worded pages. Having said that, I believe Camus’ position is often misrepresented and Cruikshank appears to agree, so a large portion of his text deals with disabusing us of the widely held belief that Camus’ entire philosophical position was that life is absurd and we have no choice but to accept it. Many cite his treatment of the Myth of Sisyphus as evidence of this position (refer next paragraph). In reality, Camus seems to have spent his entire intellectual life looking for a resolution to the absurd condition, but I’m getting ahead of myself.

 

For those who don’t know, The Myth of Sisyphus is Camus’ literary essay on a Greek mythology concerning the punishment given to Sisyphus: ‘who was condemned to repeat forever the same meaningless task of pushing a boulder up a mountain, only to see it roll down again just as it nears the top. The essay concludes, "The struggle itself towards the heights is enough to fill a man's heart. One must imagine Sisyphus happy."’ (Wikipedia)

 

Also, according to Wikipedia, ‘The absurd lies in the juxtaposition between the fundamental human need to attribute meaning to life and the "unreasonable silence" of the universe in response.’ As an aside, many scientists seem to take this position by default. In fact, Paul Davies, whom I often cite on this blog, refers to this position by most scientists as the ‘absurd universe’ without any hint of irony. In response, Davies argues that humanity is in the privileged position of being able to ‘unravel the plot’ (his phrase), but that’s perhaps too much of a diversion for this post, though I may return to it later

 

I have my own thoughts on the myth of Sisyphus in that it’s like a time loop, which of course, is not what Camus had in mind. In fact, I think the Marvel movie, Dr Strange (starring Benedict Cumberbatch) alludes to the myth towards the end, when Dr Strange traps his nemesis in a time loop – a point I make in my online review. The thing about a time loop is that you’d have no memory of the previous loop, in the same way that, if you were reincarnated, you’d have no memories of your previous incarnation. And, logically, that’s the only way one could deal with eternity.

 

I remember as quite a young child (pre-teens) lying in bed one night trying to grasp the concept of infinity or eternity and you literally look into the abyss. You realise how impossible it is to conceive – it’s a mind-fuck in the real sense of the term. In other words, we are not meant to comprehend it – it’s literally beyond our conception let alone perception. I’ve long argued that only in mathematics, does infinity have a home – in fact, it’s built into its foundations, despite the efforts of many to get rid of it.

 

So, with that in mind, the myth of Sisyphus is another way to look at eternity and show that it’s truly absurd to a human consciousness. It renders eternity into an activity that we can comprehend and viscerally feel. I know that’s not the absurdity that Camus had in mind, but that’s my take-home lesson. The truth is that we couldn’t live with immortality even if it was offered.

 

A detour. Now back to the main theme: Camus’ attempt to deal with the intrinsic absurdity of life without resorting to metaphysics. Early in his book, Cruikshank discusses a quote from The Myth of Sisyphus that brings Camus’ preoccupation with the absurd into focus.

 

“There is only one really serious philosophical problem, and that is suicide.”

 

In fact, Cruikshank discusses this at length, but basically Camus is asking us: if life is absurd, why not end it? In conclusion, Camus contends that suicide is not an escape from the absurd but giving into it, and it’s hard not to agree with him. But this begs another question: how does one live with the absurd if that’s all the universe has to offer.

 

Camus argued that we need to revolt but seemed to struggle with how that can be achieved without revolution, especially when revolutions invariably involve violence. One suspects this is a major reason for Camus rejecting Marxist communism, which led to a falling out with his contemporary French philosopher and writer, Jean-Paul Sartre. Anyone who has read Marx and Engel’s treatise on communism will know that revolution is a key step in its inception.

 

Right towards the end of Cruikshank’s book, when he discusses Camus’ plays, there is a very lucid and insightful analysis of his play, Les Justes, which is about the plotting and assassination of the Grand-Duke Sergei Alexandrovich (referred to as Grand-Duke Serge). Camus had this to say about its historical veracity:

 

However strange some situations in this play may appear they are nevertheless historically true… All my characters really existed and behaved in the way I describe. I have simply tried to give probability to what was already true.

 

Cruickshank makes the point that, in his view, this play is far superior to others Camus wrote in the depiction of the characters and the tensions between them. The reason I feel it’s worth discussing is because it seems to address the inherent tension between idealism and terrorism, personified by 2 of the main characters, that seems to arise ineluctably when people combine political ambition with violent means.

 

There is another quote by Camus in a completely different context, which I feel is relevant:

 

"But practically, I know men and recognise them by their behaviour, by the totality of their deeds, by the consequences caused in life by their presence."

 

Camus was involved in the French resistance, and would have observed the best and worst in men, as did my father in a German POW camp. They both would have appreciated better than most of us, how our moral compass can become distorted in a war environment. Both Camus, based on his writings and stated beliefs, and my father, based on what little he was willing to tell me, were very principled in the face of brutal conflict. One could argue that war is the absolute epitome of the absurd, yet men can sometimes find something abstract to hang onto so that they survive, not only physically but mentally.

 

Camus rejected existentialism, according to his own testimony, yet the one strand that runs through the novels I read and the one movie I saw, Far From Men (based on a short story) is the protagonists’ authenticity, though I suspect Camus would have loathed that term. Cruickshank repeatedly makes the point that Camus was a product of 2 environments: European France and Mediterranean Algiers. In both, I suspect, he felt an outsider and that’s a recurring theme in the fiction I am familiar with.

 

I will end by going off on a completely different tack, as is my wont. Early in his text, Cruikshank said something that caught my attention, which was an allusion to ‘truth’. This is the quote out of context:

 

Camus claims that reason is powerless and he offers no comparable alternative to truth.

 

Now, one assumes that Camus is talking about metaphysical truth or the absolute truths that religious texts claim they provide. After all, I believe this is Camus’ biggest bone of contention: that metaphysics in the form of so-called religious truths can’t deliver us from our absurd condition.

 

However, I have a counter argument. The obvious answer to me is mathematics. What’s more, I claim that mathematical truths transcend the universe therefore are metaphysical. There are 2 dictionary definitions of ‘metaphysical’, both relevant to mathematics in my Platonist view.

 

1.     Based on abstract reasoning

2.     Transcending physical matter or the laws of nature

 

Here’s the thing: given that the Universe follows mathematical rules that transcend the universe, at both its deepest level and its cosmological scale, how can it be intrinsically absurd? What would be absurd, is a universe without consciousness. It’s our very presence that makes the Universe not absurd, and I feel that’s what Camus missed, though it was staring him in the face.

 

Since I started writing fiction, I realised that nearly all fiction is about relationships. There are exceptions – Hemingway’s The Old Man and the Sea springs to mind. The subtext of all the fiction I’ve written is about relationships and companionship; and all that entails, including separation, betrayal and redemption. It’s in relationships that we find meaning in this short interval of existence, with or without metaphysical purpose.

Mathematics, language and reality

I recently read an online article with Quanta Magazine, titled How Writing Changes Mathematical Thought, featuring David E Dunning, ‘a historian of mathematics at the Smithsonian’s National Museum of American History’, who was interviewed by John Pavlus.

 

In particular, Dunning pointed out how the notation we use affects the way we explore mathematics and even comprehend it. The most significant innovation was the introduction of Hindu-Arabic numerals, along with its corresponding arithmetic, which we owe to Fibonacci (of Fibonacci numbers fame) in the 12^th Century. Tibees gives a good summary in this short video. The thing is that we would really struggle to do modern mathematics using Roman numerals, and it would be impossible for computers.

 

Dunning gives the example of the difference between Newton’s and Leibniz’s notation for calculus and how “Leibniz’s calculus got used a lot more in continental Europe, and it just grew and was fertile in a way that Newton’s wasn’t.” Which is why we all use Leibniz’s notation today.

 

But there is a more fundamental point, I believe, that Dunning doesn’t discuss. And that is the Wittgensteinian (new word) principle that the language we use limits what we can think about, because we all think in a language. And also, it’s the language of mathematics that I believe resolves the argument going back to Plato and Aristotle, whether mathematics is invented or discovered. On that last point, we invent the language but the relationships that the language describes are discovered. I contend there is a tendency to conflate the language of mathematics with mathematical formulations, because we learn them in tandem.

 

I pointed out in a much earlier post that there is also a tendency to treat mathematics as just another language, like the ones we think in, which takes the conflation I mention above to another level. The fact is that we still use the language we think in to describe mathematical notation and relationships. In other words, we absorb the language we use to do mathematics into our thinking language as a subset thereof. And this brings me back to Wittgenstein’s point, because we keep expanding our language to capture new concepts and ideas, otherwise we cognitively stagnate. And I see mathematical language as such an expansion, otherwise we can’t understand the concepts it’s describing. And perhaps this is why so many people struggle with mathematics in school, but that’s another topic.

 

One of Pavlus’s questions was: Why don’t we teach people to do math with, say, a more pictorial or visual kind of notation?

 

This is what led Dunning to talk about Newton’s and Leibniz’s respective calculus notation, but it got me thinking in a different direction.

 

Specifically, how we are visual creatures, and how I try to visualise mathematical concepts as much as possible. A graph can tell you so much more than the written equation can, and makes some concepts very easy to grasp. The best example that most people would be familiar with is a sine wave. You can see where the wave is zero and where it’s 1 and -1, and everything in between, and how it cycles in periods of 2π radians. It also shows just by looking at the graph how the cosine of an angle is 90 degrees (π/2 radians) out of phase with the corresponding sine wave, just by depicting them on the same graph.

 

Another example most of us are familiar with is a parabola being the graphical representation of a quadratic equation. The zeros (or square roots) are where the graph crosses the x axis, which can’t be greater than 2, so can have 2 square roots. However, you can have one square root if the parabola kisses the x axis and no roots if it doesn’t touch it. Though we all know we can have imaginary roots (√-1), but you need another graph which includes an imaginary axis along with the real axis.

 

In fact, complex algebra is a lot easier to understand if it’s depicted graphically. I’m a little annoyed that it wasn’t taught to me that way when I first encountered it. By depicting it on an Argand diagram, where the imaginary (i) axis replaces the y axis in a Cartesian diagram, and using polar co-ordinates, you can see how multiplication requires adding the angles, and multiplying a complex number by i means rotating everything anticlockwise by 90 degrees.

 

Even esoteric topics like Riemann’s hypothesis becomes amenable to comprehension by mortals when it’s demonstrated graphically, as this video demonstrates quite effectively.

 

Calculus is taught using graphs: the tangent of a curve being found by differentiation and the area under a curve being found by integration. Why one is the inverse function of the other, I’m not sure anyone can tell you. Differential calculus allows one to grasp the concept of instantaneity, which doesn’t physically exist, but it’s an idealism that is more than useful. Likewise, it’s almost incomprehensible that an infinite number of infinitesimal strips can give you a finite area under a curve, but it works. Calculus is like magic.

 

But I extend this visualisation into physics, where everything is depicted in the language of mathematics.

 

I never understood Einstein’s General Theory of Relativity (GR), which is a theory of gravity, until I grasped the concept of a geodesic, which can be visualised. And I can thank Richard Feynman for explaining it relatively succinctly, including mathematical formulations, in his excellent book, Six Not-So-Easy Pieces. A geodesic is the shortest distance between 2 points, and on a sphere, it’s always a great circle. Intercontinental aircraft fly along geodesics for that very reason, though they appear curved when the map is projected onto a flat surface.

 

But here’s the thing, as pointed out by Feynman: “In a uniform gravitational field the trajectory with maximum proper time for a fixed elapsed time is a parabola.” I’ll describe what he means by ‘maximum proper time’ in a moment, because that’s the key to understanding it. But we all learned that a projectile travels through the air following a parabolic curve in high school physics, without knowing anything about GR. We did it using Newton’s equations. But Einstein gives us the same result, assuming the object is not travelling at relativistic speeds.

 

And here’s why, again quoting Feynman: An object always moves from one place to another so that a clock carried on it gives a longer time than any other trajectory (italics in the original). In his words, The time measured by a moving clock is called its “proper time” (τ). In free fall, the trajectory makes the proper time of an object a maximum. And that’s what’s called a geodesic in GR.

 

And that paragraph allowed me to finally comprehend General Relativity. Any deviation of an object from free fall in a gravitational field (from its geodesic), and remember there is a gravitational field everywhere in the Universe, means its clock will slow down which is what SR (special theory of relativity) tells us. I’ve always believed that SR is dependent on GR and not the other way round, and Feynman indirectly confirmed this for me.

 

But visualisations can be misleading, and I think the wavefunction (Ψ) in Schrodinger’s equation is a case-in-point, because it’s not a physical wave. It exists in Hilbert space which, in principle, can have infinite dimensions. There is another way of expressing the same quantum mechanical (QM) phenomena and that is with Heisenberg’s matrix formulation. In fact, Heisenberg’s formulation preceded Schrodinger’s but they are mathematically equivalent. And this brings me back to Dunning’s point that the language we mathematically express something in, will give an intuitively different picture.

 

I recently read an article on Heisenberg’s revolutionary discoveries in Philosophy Now (Issue 172, Feb/Mar 2026, by Dr Kanan Purkayastha), which made the point that ‘Heisenberg attempted to calculate the behaviour of electrons around atoms using quantities we can observe’, so basically an epistemological approach. On the other hand, Schrodinger started with a principle postulated by De Broglie that an electron’s momentum could be formulated as a wave, similar to a photon, which I would call an ontological approach. Philip Ball in his book, Beyond Weird, made a similar point: that Heisenberg’s matrix approach is ‘epistemic’ and Schrodinger’s wave function approach is ‘ontic’ (his terms).

 

Many people originally thought that the famous Heisenberg Uncertainty Principle was an epistemological one, including Einstein, who said it was “just an expression of the limits of what can be determined by measurements. Or in philosophers’ terms, the nature of uncertainty would be an epistemic one.”

 

However, it falls out of Schrodinger’s equation by using a Fourier transform, so it is a mathematical restraint, not just a physical one. Schrodinger’s wavefunction also entails superposition and entanglement, which led Schrodinger to state that entanglement is the defining feature of quantum mechanics, meaning it’s what separates it from classical physics. The other thing about Schrodinger’s equation is that it can only give us probabilities, and following an observation, it no longer applies. This leads me to argue that the wavefunction exists in the future; as far as I know, an idea not shared by anyone else except Freeman Dyson (who is no longer with us).

 

Probabilities were the subject of a recent post, but the thing is we only apply probabilities to things that are yet to happen. After something has happened its probability is no longer relevant; it effectively becomes 1. And this is what happens in QM, as described above. To quote from another online article by Phys Org:

The results showed that the photon's physical presence was distributed across both paths simultaneously, demonstrating that the particle is truly delocalized until a detector forces it into a single location.

 

This is identical to a description provided by Alain Aspect that I reported in a not-so-recent post. But, as Freeman Dyson explains, it corresponds to a change in perspective by the observer from the future to the past, which occurs at the time of ‘detection’.

 

I’d like to make a point about the fact that probabilities exist, not only in QM but classical physics – after all, the entire gambling industry is based on probabilities. I contend that it means the Universe is not deterministic. Simplistic, yes, but I can’t think of a better argument. It’s also my argument against claims of so-called prophecy. You either believe in free will or you believe in prophecy, but you can’t believe in both.

 

I could imagine having a discussion (argument) with a physicist on this issue, where they claim that probabilities are a statistical outcome, as a consequence of what we cannot know. Therefore, the outcome of a coin toss, for example, could be deterministic and the probability is a consequence of our ignorance, not the event. In fact, I had this discussion (over coin tosses) with physicist, Mark John Fernee (Qld Uni). Chaos theory mathematically ensures it can never be known definitively, which is an epistemological argument. However, I argue that chaos occurs ontologically as well, and that the entire universe’s evolvement is dependent on this principle.

 

Just as in the case with Heisenberg’s Uncertainty Principle and people thinking it was a consequence of what we can't physically measure, many physicists argue that chaos theory is a consequence of our limitations of observation. However, I argue that in both cases, the limitation is built into the mathematics, which makes it a feature of the Universe.

 

So, I’ve gone way off track, but while we need a language to understand and express the mathematics we discover, nature is already determined by the rules that mathematics dictates.