Paul P. Mealing

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Showing posts with label Mathematics. Show all posts
Showing posts with label Mathematics. Show all posts

Sunday 3 March 2024

Is randomness real or illusion?

 Let’s look at quantum mechanics (QM). I watched a YouTube video on Closer To Truth with Fred Alan Wolf, a theoretical physicist, whom I admit I’d never heard of. It’s worth watching the first 7 mins before he goes off on a speculative tangent that maybe dreams are a more fundamental level of reality, citing Australian Aboriginal ‘dreamtime’ mythology, of which I have some familiarity, though no scholarship.
 
In the first 7 mins he describes QM: its conceptual frustrations juxtaposed with its phenomenal successes. He gives a good synopsis, explaining how it describes a world we don’t actually experience, yet apparently underpins (my term, not his) the one we do. In particular, he explains:
 
There is a simple operation that takes you out of that space into (hits the table with his hand) this space. And that operation is simply multiplying what that stuff - that funny stuff – is, by itself (waves his hands in circles) in a time-reverse manner, called psi star psi (Ψ*Ψ) in the language of quantum physics.
 
What he’s describing is called the Born rule, which gives probabilities of finding that ‘stuff’ in the real world. By ‘real world’ I mean the one we are all familiar with and that he can hit his hand with. Ψ (pronounced sy) is of course the wave function in Schrodinger’s eponymous equation, and Schrodinger himself wrote a paper (in 1941) demonstrating that Born’s rule effectively multiplies the wave function by itself running backwards in time.
 
Now, some physicists argue that Ψ is just a convenient mathematical fiction and Carlo Rovelli went so far as to argue that it has led us astray (in one of his popular books). Personally, I think it describes the future, which explains why we never see it, or as soon as we try to, it disappears, and if we’re lucky, we get a particle or some other interaction, like a dot on a screen, all of which exist in our past. Note that everything we observe, including our own reflection in a mirror, exists in the past.
 
Wolf then goes on to speculate that the infinite possibilities we use for our calculations are perhaps the true reality. In his own words: What I’m interested in are the things we can’t see… And he makes an interesting point that most people don’t know: that if we don’t take into account the things we can’t see, ‘we get the wrong answers’.
 

And this is where it gets interesting, because he’s alluding to Feynman’s sum-over-histories methodology, which takes into account all the infinite paths that the particle (as wave function) can take. In fact, the more paths that are allowed for, the more accurate the calculation. Wolf doesn’t mention Feynman, but I’m sure that’s what he’s referring to.
 
Feynman’s key insight into QM was that it obeys the least-action principle, which is mathematically expressed as a Lagrangian. It’s the ‘least-action principle’ that determines where light goes through a change in medium (like glass), obeying Fermat’s law where it takes the path of ‘least time’. It also determines the path a ball follows if you throw it into the air by following the path of ‘maximum relativistic time’. I elaborate on this in another post.
 
There is something teleological about this principle, as if the ball, particle, light, ‘knows’ where it has to go. Freeman Dyson, who was a close collaborator with Feynman, argued that QM cannot describe the past, but only the future, and that only classical physics describes the past. So these infinitude of paths that are part of the calculation to determine the probability of where it will actually be ‘observed’ make more sense to me if they exist in the future. I don’t think we need a ‘dream state’ unless that’s a euphemism for the future.
 
Like Dyson, I don’t think we need consciousness to make a quantum phenomenon become real, but it does provide the reference point. In his own words:
 
We do not need a human observer to make quantum mechanics work. All we need is a point of reference, to separate past from future, to separate what has happened from what may happen, to separate facts from probabilities.
 
The thing about consciousness is that it exists in a ‘constant present’, as pointed out by Schrodinger himself (when he wasn’t talking about QM), so it logically correlates with 'a point of reference, to separate past from future', that Dyson refers to.
 
Schrodinger coined a term, ‘statistico-deterministic’, to describe quantum phenomena, because, at a statistical level, it can be very predictable, otherwise we wouldn’t be able to call it ‘successful’. He gives the example of radioactive decay (exploited in his eponymous cat thought experiment) whereby we can’t determine the decay of a single isotope, yet we can statistically determine the half-life of astronomical numbers of atoms very accurately, as everyone knows.
 
I contend that real randomness, that we all observe and are familiar with, is caused by chaos, but even this is a contentious idea. I like to give the example of tossing a coin, but a lot of physicists will tell you that tossing a coin is not random. In fact, I recently had a lengthy, but respectful, discourse with Mark John Fernee (physicist at Qld Uni) on Quora on this very topic. When I raised the specific issue of whether tossing a coin is ‘random’, he effectively argued that there are no random phenomena in physics. To quote him out of context:
 
Probability theory is built from statistical sampling. There is no assumed underlying physics.
 
The underlying physics can be deterministic, while a statistical distribution of events can indicate random behaviour. This is the assumption that is applied to every coin toss. Because this is just an assumption, you can cheat the system by using specific conditions that ensure deterministic outcomes.
 
What I am saying is that randomness is a statistical characterisation of outcomes that does not include any physical mechanism. As such, it is not a fundamental property of nature.
(Emphasis in original)
 
I get the impression from what I’ve read that mathematicians have a different take on chaos to physicists, because they point out that you need to calculate initial conditions to infinite decimal places to achieve a 100% predicted outcome. Physicist, Paul Davies, provided a worked example in his 1988 book, The Cosmic Blueprint. I quoted Davies to Fernee during our ‘written’ conversation:
 
It is actually possible to prove that the activity of the jumping particle is every bit as random as tossing a coin.
 
The ‘jumping particle’ Davies referred to was an algorithm using clock arithmetic, that when graphed produced chaotic results, and he demonstrated that it would take a calculation to infinity to get it ‘exactly right’. Fernee was dismissive of this and gave it as an example of a popular science book leading laypeople (like myself) astray, which I thought was a bit harsh, as Davies actually goes into the mathematics in some detail, and I possibly misled Fernee by quoting just one sentence.
 
Just to be clear, Fernee doesn’t disagree that chaotic phenomena are impossible to predict; just that they are fully deterministic and, in his words, only ‘indicate random behaviour’.
 
Sabine Hossenfelder, who argues very strongly for superdeterminism, has a video demonstrating how predicting chaotic phenomena (like the weather) has a horizon (my term, not hers) of predictability that can never be exceeded, even in principle (10 days in the case of the weather).
 
So Fernee and Hossenfelder distinguish between what we ‘cannot know’ and what physically transpires. But my point is that chaotic phenomena, if rerun, will always produce a different result – it’s built into the mathematics underlying the activity – and includes significant life-changing phenomena like evolutionary biology and the orbits of the planets, as well as weather and earthquakes. Even the creation of the moon is believed to be a consequence of a chaotic event, without which life on Earth would never have evolved.
 
Note that both QM and chaos have mathematical underpinnings, and whilst most see that as modelling or a very convenient method of making predictions, I see it as more fundamental. I contend that mathematics transcends the Universe, yet it’s also a code that allows us to plumb Nature’s deepest secrets and fathom the dynamics of the Universe on all scales.

Sunday 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).

Sunday 31 December 2023

What are the limits of knowledge?

 This was the Question of the Month in Philosophy Now (Issue 157, August/September 2023) and 11 answers were published in Issue 159, December 2023/January 2024, including mine, which I now post complete with minor edits.

 

Some people think that language determines the limits of knowledge, yet it merely describes what we know rather than limits it, and humans have always had the facility to create new language to depict new knowledge.

There are many types of knowledge, but I’m going to restrict myself to knowledge of the natural world. The ancient Greeks were possibly the first to intuit that the natural world had its own code. The Pythagoreans appreciated that musical pitch had a mathematical relationship, and that some geometrical figures contained numerical ratios. They made the giant conceptual leap that this could possibly be a key to understanding the Cosmos itself.

Jump forward two millennia, and their insight has borne more fruit than they could possibly have imagined. Richard Feynman made the following observation about mathematics in The Character of Physical Law: “Physicists cannot make a conversation in any other language. If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. She offers her information only in one form.”

Meanwhile, the twentieth century logician Kurt Gödel proved that in any self-consistent, axiom-based, formal mathematical system, there will always be mathematical truths that can’t be proved true using that system. However, they potentially can be proved if one expands the axioms of the system. This infers that there is no limit to mathematical truths.

Alonso Church’s ‘paradox of unknowability’ states, “unless you know it all, there will always be truths that are by their very nature unknowable.” This applies to the physical universe itself. Specifically, since the vast majority of the Universe is unobservable, and possibly infinite in extent, most of it will remain forever unknowable. Given that the limits of knowledge are either infinite or unknowable in both the mathematical and physical worlds, then those limits are like a horizon that retreats as we advance towards it.

Friday 17 November 2023

On the philosophy of reality

 This follows on from my last post, after I saw a YouTube interview with Raymond Tallis on Closer to Truth. He’s all but saying that physics has lost the plot, or at least that’s my takeaway. I happen to know that he’s also writing a book on ‘reality’ – might even have finished it – which is why he can’t stop talking about it, and, it seems, neither can I.
 
I think there are 3 aspects to this discussion, even though they are not clearly delineated. Nevertheless, it might be worth watching the video to better appreciate what I’m talking about. While I agree with some of his points, I think Tallis’s main thrust that physicists contend that ‘reality dissolves’ is a strawman argument as I’ve never heard or read a physicist make that claim. Robert Lawrence Kuhn, who hosts all the talks on Closer To Truth, appears to get uncharacteristically flustered, but I suspect it’s because he intuitively thought the argument facile but couldn’t easily counter it. It would have been far more interesting and edifying if Tallis was debating with someone like Paul Davies, who is not only a physicist, but knows some philosophy.
 
At one point they get onto evolution, as Kuhn attempts to make the distinction between how we’ve evolved to understand the world but culturally moved beyond that. This leads to the 3 aspects I alluded to earlier.
 
The first aspect is that there is an objective reality independent of us, which we need to take seriously because it can kill us. As Tallis points out, this is what we’ve evolved to avoid, otherwise we wouldn’t be here. As I’ve pointed out many times, our brains create a model of that reality so we can interact with it. This is the second aspect, and is part of our evolutionary heritage.
 
The third aspect appears to be completely at odds with this and that appears to be what Tallis has an issue with. The third aspect is that we make mathematical models of reality, which seem, on the surface at least, to have no bearing on the reality that we experience. We don’t see wavefunctions of particles or twins aging at different rates when one goes on a journey somewhere.
 
It doesn’t help that different physicists attempt to give different accounts of what’s happening. For example, a lot of physicists believe that the wavefunction is just a useful mathematical fiction. Others believe that it carries on in another universe after the ‘observation’ or ‘measurement’. All acknowledge that we can’t explain exactly what happens, which is why it’s called the ‘measurement problem’.
 
What many people don’t tell you is that QM only makes predictions about events, which is why it deals in probabilities, and logically, observations require a time lapse, no matter how small, before it’s recorded, so it axiomatically happens in the past. As Paul Davies points out there is an irreversibility in time once the ‘observation’ has been made.
 
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.
 
So, nothing ‘dissolves’, it’s just not observable until after the event, and the event could be a photon hitting a photo-sensitive surface or an isotope undergoing some form of radioactive decay or an electron hitting a screen and emitting light. Even Sabine Hossenfleder (in one of her videos) points out that the multiple paths of Feynman’s ‘sum-over-histories path-integral’ are in the future of the measurement that they predict via calculation.
 
Tallis apparently thinks that QM infers that there is nothing solid in the world, yet it was Freeman Dyson, in collaboration with Andrew Leonard, who used Wolfgang Pauli’s Exclusion Principle to demonstrate why solid objects don’t meld into each other. Dyson acknowledged that ‘the proof was extraordinarily complicated, difficult and opaque’, which might explain why it took so long for someone to calculate it (1967).
 
Humans are unique within the animal kingdom in that we’ve developed tools that allow us to ‘sense’ phenomena that can’t be detected through our biological senses. It’s this very attribute that has led to the discipline of science, and in the last century it has taken giant strides beyond anything our predecessors could have imagined. Not only have we learned that we live in a galaxy that is one among trillions and that the Universe is roughly 14 billion years old, but we can ‘sense’ radiation only 380,000 years after its birth. Who would have thought? At the other end of the scale, we’ve built a giant underground synchrotron that ‘senses’ the smallest known particle in nature, called quarks. They are sub-sub-atomic.
 
But, in conjunction with these miracle technologies, we have discovered, or developed (a combination of both), mathematical tools that allow us to describe these phenomena. In fact, as Richard Feynman pointed out, mathematics is the only language in which ‘nature speaks’. It’s like the mathematical models are another tool in addition to the technological ones that extend our natural senses.
 
Having said that, sometimes these mathematical models don’t actually reflect the real world. A good example is Ptolemy’s model of the solar system using epicycles, that had Earth at its centre. A possible modern example is String Theory, which predicts up to 10 spatial dimensions when we are only aware of 3.
 
Sabine Hossenfelder (already mentioned) wrote a book called Lost in Math, where she challenges this paradigm. I think that this is where Tallis is coming from, though he doesn’t specifically say so. He mentions a wavefunction (in passing), and I’ve already pointed out that some physicists see it as a convenient and useful mathematical fiction. One is Viktor T Toth (on Quora) who says:
 
The mathematical fiction of wavefunction collapse was “invented” to deal with the inconvenient fact that otherwise, we’d have to accept what the equations tell us, namely that quantum mechanics is nonlocal (as per Bell’s theorem)…

 
But it’s this very ‘wavefunction collapse’ that Davies was referring to when he pointed out that it ‘destroys the rewind button’. Toth has a different perspective:
 
As others pointed out, wavefunction collapse is, first and foremost, a mathematical abstraction, not a physical process. If it were a physical process, it would be even weirder. Rather than subdividing spacetime with an arbitrarily chosen hypersurface called “now” into a “before observation” and an “after observation” half, connected by the non-unitary transformation of the “collapse”, wavefunction collapse basically implies throwing away the entire universe, replacing it with a different one (past, present, and future included) containing the collapsed wavefunction instead of the original.
 
Most likely, it’s expositions like this that make Tallis throw up his hands (figuratively speaking), even though I expect he’s never read anything by Toth. Just to address Toth’s remark, I would contend that the ‘arbitrarily chosen hypersurface called “now”’ is actually the edge in time of the entire universe. A conundrum that is rarely acknowledged, let alone addressed, is that the Universe appears to have no edge in space while having an edge in time. Notice how different his ‘visualisation’ is to Davies’, yet both of them are highly qualified and respected physicists.
 
So, while there are philosophical differences among physicists, one can possibly empathise with the frustrations of a self-identified philosopher. (Tallis’s professional background is in neuroscience.)
 
Nevertheless, Tallis uses quantum mechanics just like the rest of us, because all electronic devices are dependent on it, and we all exploit Einstein’s relativity theories when we use our smartphones to tell us where we are.
 
So the mathematical models, by and large, work. And they work so well, that we don’t need to know anything about them, in the same way you don’t need to know anything about all the technology your car uses in order for you to drive it.
 
Tallis, like many philosophers, sees mathematics as a consequence of our ability to measure things, which we then turn into equations that conveniently describe natural phenomena. But the history of Western science reveals a different story, where highly abstract mathematical discoveries later provide an epistemological key to our comprehension of the most esoteric natural phenomena. The wavefunction is a good example: using an unexpected mathematical relationship discovered by Euler in the 1700s, it encapsulates in one formula (Shrodinger’s), superposition, entanglement and Heisenberg’s Uncertainty Principle. So it may just be a mathematical abstraction, yet it describes the most enigmatic features discovered in the natural world thus far.
 
From what I read and watch (on YouTube), I don’t think you can do theoretical physics without doing philosophy. Philosophy (specifically, epistemology) looks at questions that don’t have answers using our current bank of knowledge. Examples include the multiverse, determinism and free will. Philosophers with a limited knowledge of physics (and that includes me) are not in the same position as practicing physicists to address questions about reality. This puts Tallis at a disadvantage. Physicists can’t agree on topics like the multiverse, superdeterminism, free will or the anthropic principle, yet often hold strong views regardless.
 
I’m always reminded of John Wheeler’s metaphor of science as an island of knowledge in a sea of ignorance, with the shoreline being philosophy. Note that as the island expands so does the shoreline of our ignorance.

Monday 23 October 2023

The mystery of reality

Many will say, ‘What mystery? Surely, reality just is.’ So, where to start? I’ll start with an essay by Raymond Tallis, who has a regular column in Philosophy Now called, Tallis in Wonderland – sometimes contentious, often provocative, always thought-expanding. His latest in Issue 157, Aug/Sep 2023 (new one must be due) is called Reflections on Reality, and it’s all of the above.
 
I’ve written on this topic many times before, so I’m sure to repeat myself. But Tallis’s essay, I felt, deserved both consideration and a response, partly because he starts with the one aspect of reality that we hardly ever ponder, which is doubting its existence.
 
Actually, not so much its existence, but whether our senses fool us, which they sometimes do, like when we dream (a point Tallis makes himself). And this brings me to the first point about reality that no one ever seems to discuss, and that is its dependence on consciousness, because when you’re unconscious, reality ceases to exist, for You. Now, you might argue that you’re unconscious when you dream, but I disagree; it’s just that your consciousness is misled. The point is that we sometimes remember our dreams, and I can’t see how that’s possible unless there is consciousness involved. If you think about it, everything you remember was laid down by a conscious thought or experience.
 
So, just to be clear, I’m not saying that the objective material world ceases to exist without consciousness – a philosophical position called idealism (advocated by Donald Hoffman) – but that the material objective world is ‘unknown’ and, to all intents and purposes, might as well not exist if it’s unperceived by conscious agents (like us). Try to imagine the Universe if no one observed it. It’s impossible, because the word, ‘imagine’, axiomatically requires a conscious agent.
 
Tallis proffers a quote from celebrated sci-fi author, Philip K Dick: 'Reality is that which, when you stop believing in it, doesn’t go away' (from The Shifting Realities of Philip K Dick, 1955). And this allows me to segue into the world of fiction, which Tallis doesn’t really discuss, but it’s another arena where we willingly ‘suspend disbelief' to temporarily and deliberately conflate reality with non-reality. This is something I have in common with Dick, because we have both created imaginary worlds that are more than distorted versions of the reality we experience every day; they’re entirely new worlds that no one has ever experienced in real life. But Dick’s aphorism expresses this succinctly. The so-called reality of these worlds, in these stories, only exist while we believe in them.
 
I’ve discussed elsewhere how the brain (not just human but animal brains, generally) creates a model of reality that is so ‘realistic’, we actually believe it exists outside our head.
 
I recently had a cataract operation, which was most illuminating when I took the bandage off, because my vision in that eye was so distorted, it made me feel sea sick. Everything had a lean to it and it really did feel like I was looking through a lens; I thought they had botched the operation. With both eyes open, it looked like objects were peeling apart. So I put a new eye patch on, and distracted myself for an hour by doing a Sudoku problem. When I had finished it, I took the patch off and my vision was restored. The brain had made the necessary adjustments to restore the illusion of reality as I normally interacted with it. And that’s the key point: the brain creates a model so accurately, integrating all our senses, but especially, sight, sound and touch, that we think the model is the reality. And all creatures have evolved that facility simply so they can survive; it’s a matter of life-and-death.
 
But having said all that, there are some aspects of reality that really do only exist in your mind, and not ‘out there’. Colour is the most obvious, but so is sound and smell, which all may be experienced differently by other species – how are we to know? Actually, we do know that some animals can hear sounds that we can’t and see colours that we don’t, and vice versa. And I contend that these sensory experiences are among the attributes that keep us distinct from AI.
 
Tallis makes a passing reference to Kant, who argued that space and time are also aspects of reality that are produced by the mind. I have always struggled to understand how Kant got that so wrong. Mind you, he lived more than a century before Einstein all-but proved that space and time are fundamental parameters of the Universe. Nevertheless, there are more than a few physicists who argue that the ‘flow of time’ is a purely psychological phenomenon. They may be right (but arguably for different reasons). If consciousness exists in a constant present (as expounded by Schrodinger) and everything else becomes the past as soon as it happens, then the flow of time is guaranteed for any entity with consciousness. However, many physicists (like Sabine Hossenfelder), if not most, argue that there is no ‘now’ – it’s an illusion.
 
Speaking of Schrodinger, he pointed out that there are fundamental differences between how we sense sight and sound, even though they are both waves. In the case of colour, we can blend them to get a new colour, and in fact, as we all know, all the colours we can see can be generated by just 3 colours, which is how the screens on all your devices work. However, that’s not the case with sound, otherwise we wouldn’t be able to distinguish all the different instruments in an orchestra. Just think: all the complexity is generated by a vibrating membrane (in the case of a speaker) and somehow our hearing separates it all. Of course, it can be done mathematically with a Fourier transform, but I don’t think that’s how our brains work, though I could be wrong.
 
And this leads me to discuss the role of science, and how it challenges our everyday experience of reality. Not surprisingly, Tallis also took his discussion in that direction. Quantum mechanics (QM) is the logical starting point, and Tallis references Bohr’s Copenhagen interpretation, ‘the view that the world has no definite state in the absence of observation.’ Now, I happen to think that there is a logical explanation for this, though I’m not sure anyone else agrees. If we go back to Schrodinger again, but this time his eponymous equation, it describes events before the ‘observation’ takes place, albeit with probabilities. What’s more, all the weird aspects of QM, like the Uncertainty Principle, superposition and entanglement, are all mathematically entailed in that equation. What’s missing is relativity theory, which has since been incorporated into QED or QFT.
 
But here’s the thing: once an observation or ‘measurement’ has taken place, Schrodinger’s equation no longer applies. In other words, you can’t use Schrodinger’s equation to describe something that has already happened. This is known as the ‘measurement problem’, because no one can explain it. But if QM only describes things that are yet to happen, then all the weird aspects aren’t so weird.
 
Tallis also mentions Einstein’s 'block universe', which infers past, present and future all exist simultaneously. In fact, that’s what Sabine Hossenfelder says in her book, Existential Physics:
 
The idea that the past and future exist in the same way as the present is compatible with all we currently know.

 
And:

Once you agree that anything exists now elsewhere, even though you see it only later, you are forced to accept that everything in the universe exists now. (Her emphasis.)
 
I’m not sure how she resolves this with cosmological history, but it does explain why she believes in superdeterminism (meaning the future is fixed), which axiomatically leads to her other strongly held belief that free will is an illusion; but so did Einstein, so she’s in good company.
 
In a passing remark, Tallis says, ‘science is entirely based on measurement’. I know from other essays that Tallis has written, that he believes the entire edifice of mathematics only exists because we can measure things, which we then applied to the natural world, which is why we have so-called ‘natural laws’. I’ve discussed his ideas on this elsewhere, but I think he has it back-to-front, whilst acknowledging that our ability to measure things, which is an extension of counting, is how humanity was introduced to mathematics. In fact, the ancient Greeks put geometry above arithmetic because it’s so physical. This is why there were no negative numbers in their mathematics, because the idea of a negative volume or area made no sense.
 
But, in the intervening 2 millennia, mathematics took on a life of its own, with such exotic entities like negative square roots and non-Euclidean geometry, which in turn suddenly found an unexpected home in QM and relativity theory respectively. All of a sudden, mathematics was informing us about reality before measurements were even made. Take Schrodinger’s wavefunction, which lies at the heart of his equation, and can’t be measured because it only exists in the future, assuming what I said above is correct.
 
But I think Tallis has a point, and I would argue that consciousness can’t be measured, which is why it might remain inexplicable to science, correlation with brain waves and their like notwithstanding.
 
So what is the mystery? Well, there’s more than one. For a start there is consciousness, without which reality would not be perceived or even be known, which seems to me to be pretty fundamental. Then there are the aspects of reality which have only recently been discovered, like the fact that time and space can have different ‘measurements’ dependent on the observer’s frame of reference. Then there is the increasing role of mathematics in our comprehension of reality at scales both cosmic and subatomic. In fact, given the role of numbers and mathematical relationships in determining fundamental constants and natural laws of the Universe, it would seem that mathematics is an inherent facet of reality.
 

Sunday 10 September 2023

A philosophical school of thought with a 2500 year legacy

I’ve written about this before, but revisited it with a recent post I published on Quora in response to a question, where I didn’t provide the answer expected, but ended up giving a very brief history of philosophy as seen through the lens of science.
 
I’ve long contended that philosophy and science are joined at the hip, and one might extend the metaphor by saying the metaphysical bond is mathematics.
 
When I say a very brief history, what I mean is that I have selected a few specific figures, albeit historically prominent, who provide links in a 2500 year chain, while leaving out countless others. I explain how I see this as a ‘school of thought’, analogous to how some people might see a religion that also goes back centuries. The point is that we in the West have inherited this, and it’s determined the technological world that we currently live in, which would have been unimaginable even as recently as the renaissance or the industrial revolution, let alone in ancient Greece or Alexandria.
 
Which philosopher can you best relate yourself to?
 
It would take a certain hubris to claim that I relate to any philosopher whom I admire, but there are some whom I feel, not so much a kinship with, but an agreement in spirit and principle. Philosophers, like scientists and mathematicians, stand on the shoulders of those who went before.
 
I go back to Socrates because I think he was ahead of his time, and he effectively brought argument into philosophy, which is what separates it from dogma.
 
Plato was so influenced by Socrates that he gave us the ‘Socratic dialogue’ method of analysing an issue, whereby fictional characters (albeit with historical names) discuss hypotheticals in the form of arguments.
 
But Plato was also heavily influenced by Pythagorean philosophy, and even adopted its quadrivium of arithmetic, geometry, astronomy and music for his famous Academy. This tradition was carried over to the famous school or Library of Alexandria, from which sprang such luminaries as Euclid, Eratosthenes, who famously ‘measured’ the circumference of the Earth (around 230BC) and Hypatia, the female mathematician, mentor to a Bishop and a Roman Prefect, as well as speaker in the Senate, who was killed for her sins by a Christian mob in 414AD.
 
Plato is most famously known for his cave allegory, whereby we observe shadows on a wall, without knowing that there is another reality beyond our kin, consequently called the Platonic realm. In later years, this was associated with the Christian ideal of ‘heaven’, but was otherwise considered an outdated notion.
 
Then, jumping forward a couple of centuries from Plato, we come to Kant, who inadvertently resurrected the idea with his concept of ‘transcendental idealism’. Kant famously postulated that there is a difference between what we observe and the ‘thing-in-itself’, which we may never know. I find this reminiscent of Plato’s cave analogy.
 
Even before Kant there was a scientific revolution led by Galileo, Kepler and Newton, who took Pythagorean ideals to a new level when they used geometry and a new mathematical method called calculus to describe the motions of the planets that had otherwise escaped a proper and consistent exposition.
 
Then came the golden age of physics that not only built on Newton, but also Faraday and Maxwell, whereby newly discovered mathematical tools like complex algebra and non-Euclidean geometry opened up a Pandora’s box called quantum mechanics and relativity theory, which have led the way for over a hundred years in our understanding of the infinitesimally small and the cosmologically large, respectively.
 
But here’s the thing: since the start of the last century, all our foundational theories have been led by mathematics rather than experimentation, though the latter is required to validate the former.
 
To quote Richard Feynman from a chapter in his book, The Character of Physical Law, titled, The Relation of Mathematics to Physics:


Physicists cannot make a conversation in any other language. If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. She offers her information only in one form.
 
And this leads me to conclude that Kant's ‘transcendental idealism’ is mathematics*, which has its roots going back to Plato and possibly Pythagoras before him.
 
In answer to the question, I don’t think there is any specific philosopher that I ‘best relate to’, but there is a school of thought going back 2500 years that I have an affinity for.
 
 
*Note: Kant didn’t know that most of mathematics is uncomputable and unknown.
 

Thursday 25 May 2023

Philosophy’s 2 disparate strands: what can we know; how can we live

The question I’d like to ask, is there a philosophical view that encompasses both? Some may argue that Aristotle attempted that, but I’m going to take a different approach.
 
For a start, the first part can arguably be broken into 2 further strands: physics and metaphysics. And even this divide is contentious, with some arguing that metaphysics is an ‘abstract theory with no basis in reality’ (one dictionary definition).
 
I wrote an earlier post arguing that we are ‘metaphysical animals’ after discussing a book of the same name, though it was really a biography of 4 Oxford women in the 20th Century: Elizabeth Anscombe, Mary Midgley, Philippa Foot and Iris Murdoch. But I’ll start with this quote from said book.
 
Poetry, art, religion, history, literature and comedy are all metaphysical tools. They are how metaphysical animals explore, discover and describe what is real (and beautiful and good). (My emphasis.)
 
So, arguably, metaphysics could give us a connection between the 2 ‘strands’ in the title. Now here’s the thing: I contend that mathematics should be part of that list, hence part of metaphysics. And, of course, we all know that mathematics is essential to physics as an epistemology. So physics and metaphysics, in my philosophy, are linked in a rather intimate  way.
 
The curious thing about mathematics, or anything metaphysical for that matter, is that, without human consciousness, they don’t really exist, or are certainly not manifest. Everything on that list is a product of human consciousness, notwithstanding that there could be other conscious entities somewhere in the universe with the same capacity.
 
But again, I would argue that mathematics is an exception. I agree with a lot of mathematicians and physicists that while we create the symbols and language of mathematics, we don’t create the intrinsic relationships that said language describes. And furthermore, some of those relationships seem to govern the universe itself.
 
And completely relevant to the first part of this discussion, the limits of our knowledge of mathematics seems to determine the limits of our knowledge of the physical world.
 
I’ve written other posts on how to live, specifically, 3 rules for humans and How should I live? But I’m going to go via metaphysics again, specifically storytelling, because that’s something I do. Storytelling requires an inner and outer world, manifest as character and plot, which is analogous to free will and fate in the real world. Now, even these concepts are contentious, especially free will, because many scientists tell us it’s an illusion. Again, I’ve written about this many times, but it’s relevance to my approach to fiction is that I try and give my characters free will. An important part of my fiction is that the characters are independent of me. If my characters don’t take on a life of their own, then I know I’m wasting my time, and I’ll ditch that story.
 
Its relevance to ‘how to live’ is authenticity. Artists understand better than most the importance of authenticity in their work, which really means keeping themselves out of it. But authenticity has ramifications, as any existentialist will tell you. To live authentically requires an honesty to oneself that is integral to one’s being. And ‘being’ in this sense is about being human rather than its broader ontological meaning. In other words, it’s a fundamental aspect of our psychology, because it evolves and changes according to our environment and milieu. Also, in the world of fiction, it's a fundamental dynamic.
 
What's more, if you can maintain this authenticity (and it’s genuine), then you gain people’s trust, and that becomes your currency, whether in your professional life or your social life. However, there is nothing more fake than false authenticity; examples abound.
 
I’ll give the last word to Socrates; arguably the first existentialist.
 
To live with honour in this world, actually be what you try to appear to be.


Sunday 16 April 2023

From Plato to Kant to physics

 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.

Wednesday 22 March 2023

The Library of Babel

 You may have heard of this mythic place. There was an article in the same Philosophy Now magazine I referenced in my last post, titled World Wide Web or Library of Babel? By Marco Nuzzaco. Apparently, Jorge Luis Borges (1899-1986) wrote a short story, The Library of Babel in 1941. A little bit of research reveals there are layers of abstraction in this imaginary place, extrapolated upon by another book, The Unimaginable Mathematics of Borges’ Library of Babel, by Mathematical Professor, William Goldbloom Bloch, published in 2008 by Oxford University Press and receiving an ‘honourable mention’ in the 2009 PROSE Awards. I should point out that I haven’t read either of them, but the concept fascinates me, as I expound upon below.
 
The Philosophy Now article compares it with the Internet (as per the title), because the Internet is quickly becoming the most extensive collection of knowledge in the history of humanity. To quote the author, Nuzzarco:
 
The amount of information produced on the Internet in the span of 10 years from 2010 to 2020 is exponentially and incommensurably larger than all the information produced by humanity in the course of its previous history.
 
And yes, the irony is not lost on me that this blog is responsible for its own infinitesimal contribution. But another quote from the same article provides the context that I wish to explore.
 
The Library of Babel contains all the knowledge of the universe that we can possibly gain. It has always been there, and it always will be. In this sense, the knowledge of the library reflects the universe from a God’s eye perspective and the librarians’ relentless research is to decipher its secrets and its mysterious order and purpose – or maybe, as Borges wonders, the ultimate lack of any of these.

 
One can’t read this without contemplating the history of philosophy and science (at least, in the Western tradition) that has attempted to do exactly that. In fact, the whole enterprise has a distinctive Platonic flavour to it, because there is one sense in which the fictional Library of Babel is ‘real’, and it links back to my last post.
 
I haven’t read Borges’ or Bloch’s books, so I’m simply referring to the concept alluded to in that brief quote, that there is an abstract landscape or territory that humans have the unique capacity to explore. And anyone who has considered the philosophy of mathematics knows that it fulfills that criterion.
 
Mathematics has unlocked more secrets about the Universe than any other endeavour. There is a similarity here to Paul Davies’ metaphor of a ‘warehouse’ (which he expounds upon in this video) but I think a Library is an even more apposite allusion. We are like ‘librarians’ trying to decipher God’s view of the Universe that we inhabit, and to extend the metaphor, God left behind a code that only we can decipher (as far as we know) and that code is mathematics.
 
To quote Feynman (The Character of Physical Law, specifically in a chapter titled The Relation of Mathematics to Physics):
 
Physicists cannot make a conversation in any other language. If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. She offers her information only in one form.

 
And if we have the knowledge of Gods then we also have the power of Gods, and that is what we’re witnessing, right now, in our current age. We have the power to destroy the world on which we live, either in a nuclear conflagration or runaway climate change (we are literally changing the weather). But we can also use the same knowledge to make the world a more inhabitable place, but to do that we need to be less human-centric.
 
If there is a God, then (he/she) has left us in charge. I think I’ve written about that before. So yes, we are the ‘Librarians’ who have access to extraordinary knowledge and with that knowledge comes extraordinary responsibilities.

 

Friday 17 March 2023

In the beginning there was logic

 I recently read an article in Philosophy Now (Issue 154, Feb/Mar 2023), jointly written by Owen Griffith and A.C. Paseau, titled One Logic, Or Many? Apparently, they’ve written a book on this topic (One True Logic, Oxford University Press, May 2022).
 
One of the things that struck me was that they differentiate between logic and reason, because ‘reason is something we do’. This is interesting because I’ve argued previously that logic should be a verb, but I concede they have a point. In the past I saw logic as something that’s performed, by animals and machines as well as humans. And one of the reasons I took this approach was to distinguish logic from mathematics. I contend that we use logic to access mathematics via proofs, which we then call theorems. But here’s the thing: Kurt Godel proved, in effect, that there will always be mathematical ‘truths’ that we can’t prove within any formal system of mathematics that is consistent. The word ‘consistent’ is important (as someone once pointed out to me) because, if it’s inconsistent, then all bets are off.
 
What this means is that there is potentially mathematics that can’t be accessed by logic, and that’s what we’ve found, in practice, as well as in principle. Matt Parker provides a very good overview in this YouTube video on what numbers we know and what we don’t know. And what we don’t know is infinitely greater than what we do know. Gregory Chaitin has managed to prove that there are infinitely greater incomputable numbers than computable numbers, arguing that Godel’s Incompleteness Theorem goes to the very foundation of mathematics.
 
This detour is slightly off-topic, but very relevant. There was a time when people believed that mathematics was just logic, because that’s how we learned it, and certainly there is a strong relationship. Without our prodigious powers of logic, mathematics would be an unexplored territory to us, and remain forever unknown. There are even scholars today who argue that mathematics that can’t be computed is not mathematics, which rules out infinity. That’s another discussion which I won’t get into, except to say that infinity is unavoidable in mathematics. Euclid (~300 BC) proved (using very simple logic) that you can have an infinite number of primes, and primes are the atoms of arithmetic, because all other numbers can be derived therefrom.
 
The authors pose the question in their title: is there a pluralism of logic? And compare a logic relativism with moral relativism, arguing that they both require an absolutism, because moral relativism is a form of morality and logic relativism is a form of logic, neither of which are relative in themselves. In other words, they always apply by self-definition, so contradict the principle that they endorse – they are outside any set of rules of morality or logic, respectively.
 
That’s their argument. My argument is that there are tenets that always apply, like you can’t have a contradiction. They make this point themselves, but one only has to look at mathematics again. If you could allow contradictions, an extraordinary number of accepted proofs in mathematics would no longer apply, including Euclid’s proof that there are an infinity of primes. The proof starts with the premise that you have the largest prime number and then proves that it isn’t.
 
I agree with their point that reason and logic are not synonymous, because we can use reason that’s not logical. We make assumptions that can’t be confirmed and draw conclusions that rely on heuristics or past experiences, out of necessity and expediency. I wrote another post that compared analytical thinking with intuition and I don’t want to repeat myself, but all of us take mental shortcuts based on experience, and we wouldn’t function efficiently if we didn’t.
 
One of the things that the authors don’t discuss (maybe they do in their book) is that the Universe obeys rules of logic. In fact, the more we learn about the machinations of the Universe, on all scales, the more we realise that its laws are fundamentally mathematical. Galileo expressed this succinctly in the 17th Century, and Richard Feynman reiterated the exact same sentiment in the last century.
 
Cliffard A Pickover wrote an excellent book, The Paradox of God And the Science of Omniscience, where he points out that even God’s omniscience has limits. To give a very trivial example, even God doesn’t know the last digit of pi, because it doesn’t exist. What this tells me is that even God has to obey the rules of logic. Now, I’ve come across someone (Sye Ten Bruggencate) who argued that the existence of logic proves the existence of God, but I think he has it back-to-front (if God can’t breach the rules of logic). In other words, if God invented logic, ‘He’ had no choice. And God can’t make a prime number nonprime or vice versa. There are things an omnipotent God can’t do and there are things an omniscient God can’t know. So, basically, even if there is a God, logic came first, hence the title of this essay.

Tuesday 16 August 2022

How does science work?

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

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

 

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

Wednesday 10 August 2022

What is knowledge? And is it true?

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

Saturday 11 June 2022

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

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

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

 

Comment:

 

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

 

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

 

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

 

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

 

Maybe I should write this as a separate answer.

 

And I did:

 

One word answer: No.

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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


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

 

 

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


Wednesday 27 April 2022

Is infinity real?

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

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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