What’s the most fundamental value?

 This is a Question of the Month in Philosophy Now (Issue 143, April/May 2021). I wrote it very quickly, almost on impulse in less than ½ hr, but I spent a lot of time polishing it.

The word ‘fundamental’ is key here: it infers the cornerstone or foundation upon which all other values are built. Carlo Rovelli, who is better known as a physicist than a philosopher, said in an online video that “we are not entities, we are relations”. And I believe this aphorism goes to the heart of what it means to be human. From our earliest cognitive moments to the very end of our days, the quality of our lives is largely dependent on our relationships with others. And, in that context, I would contend that the most important and fundamental value is trust. Without trust, honesty does not have a foothold, and arguably honesty is the glue in any relationship, be it familial, contractual or even between governments and the general public. 

 

Psychologists will tell you that fear and trust cannot co-exist. If someone, either as a child, or a spouse, is caught in a relationship governed by fear, yet completely dependent, the consequence will inevitably result in an inability to find intimacy outside that relationship, because trust will be corroded if not destroyed. 

 

Societies can’t function without trust: traffic would be chaos; projects wouldn’t be executed collaboratively. We all undertake financial transactions every day and there is a strong element of trust involved in all of these that most of us take for granted. Cynics will argue that trust allows others to take advantage of you, which means trust only works if it is reciprocated. If enough people take advantage of those who trust, then it would evaporate and everyone would suddenly dissemble and obfuscate. Relationships would be restricted to one’s closest family and wider interactions would be fraught with hidden agendas, even paranoia. But this is exactly what happens when governments mandate their citizenry to ‘out’ people who don’t toe the party line. 

 

Everything that we value in our relationships and friendships, be it love, integrity, honesty, loyalty or respect, is forfeit without trust. As Carlo Rovelli intimated in his aphoristic declaration, it is through relationships that we are defined by others and how we define ourselves. It is through these relationships that we find love, happiness, security and a sense of belonging. We ultimately judge our lives by the relationships we form over time, both in our professional lives and our social lives. Without trust, they simply don’t exist, except as fake.

                                                --------------------------

I once wrote on this topic before, in 2008. I deliberately avoided reading that post while I wrote this one. To be honest, I’m glad I did as it’s a much better post. However, this is a response to a specific question with a limit of 400 words. Choosing the answer was the easy part – it took seconds – arguing a case was more organic. I’ll add an addendum if it’s published.

Interestingly, 'trust' crops up in my fiction more than once. In the last story I wrote, it took centre stage.

Philosophy of mathematics

 I’ve been watching a number of YouTube videos on this topic, although some of them are just podcasts with a fixed-image screen – usually a blackboard of equations. I’ll provide links to the ones I feel most relevant. I’ve discussed this topic before, but these videos have made me reassess and therefore re-analyse different perspectives. My personal prejudice is mathematical Platonism, so while I’ll discuss other philosophical positions, I won’t make any claim to neutrality.

What I’ve found is that you can divide all the various preferred views into 3 broad categories. Mathematics as abstract ‘objects’, which is effectively Platonism; mathematics as a human construct; and mathematics as a descriptive representation of the physical world. These categories remind one of Penrose’s 3 worlds, which I’ve discussed in detail elsewhere. None of the talks I viewed even mention Penrose, so henceforth, neither will I. I contend that all the various non-Platonic ‘schools’, like formalism, constructivism, logicism, nominalism, Aristotlean realism (not an exhaustive list) fall into either of these 2 camps (mental or physical attribution) or possibly a combination of both. 

 

So where to start? Why not start with numbers, as at least a couple of the videos did. We all learn numbers as children, usually by counting objects. And we quickly learned that it’s a concept independent of the objects being counted. In fact, many of us learned the concept by counting on our fingers, which is probably why base 10 arithmetic is so universal. So, in this most simple of examples, we already have a combination of the mental and the physical. I once made the comment on a previous post that humans invented numbers but we didn’t invent the relationships between them. More significantly, we didn’t specify which numbers are prime and which are non-prime – it’s a property that emerges independently of our counting or even what base arithmetic we use. I highlight primes, in particular, because they are called ‘the atoms of mathematics’, and we can even prove that they go to infinity.

 

But having said that, do numbers exist independently of the Universe? (As someone in one of the videos asked.) Ian Stewart was the first person I came across who defined ‘number’ as a concept, which infers they are mental constructs. But, as pointed out in the same video, we have numbers like pi which we can calculate but which are effectively uncountable. Even the natural numbers themselves are infinite and I believe this is the salient feature of mathematics. Anything that’s infinite transcends the Universe, almost by definition. So there will always be aspects of mathematics that will be unknowable, yet, we can ‘prove’ they exist, therefore they must exist outside of space and time. In a nutshell, that’s my best argument for mathematical Platonism.

 

But the infinite nature of mathematics means that even computers can’t deal with a completely accurate version of pi – they can only work with an approximation (as pointed out in the same video). This has led some mathematicians to argue that only computable numbers can be considered part of mathematics. Sydney based mathematician, Norman Wildberger, provides the best arguments I’ve come across for this rather unorthodox view. He claims that the Real numbers don’t exist, and is effectively a crusader for a new mathematical foundation that he believes will reinvent the entire field.

Probably the best talk I heard was a podcast from The Philosopher’s Zone, which is a regular programme on ABC Radio National, where presenter, Alan Saunders, interviewed James Franklin, Professor in the School of Mathematics and Statistics at UNSW (University of New South Wales). I would contend there is a certitude in mathematics we don't find in other fields of human endeavour. Freeman Dyson once argued that a mathematical truth is for all time – it doesn’t get overturned by subsequent discoveries.

 

And one can’t talk about mathematical ‘truth’ without talking about Godel’s Incompleteness Theorem. Godel created a self-referencing system of logic, whereby he created the mathematical equivalent of the ‘liar paradox’ – ‘this statement is false’. He effectively demonstrated that within any ‘formal’ system of mathematics you can’t prove ‘consistency’. This video by Mark Colyvan (Professor of Philosophy and Director of the Sydney Centre for the Foundations of Science), explains it better than I can. I’m not a logician, so I’m not going to expound on something I don’t fully understand, but the message I take from Godel is that he categorically showed there is a fundamental difference between ‘truth’ and ‘proof’ in mathematics. Basically, in any axiom-based mathematical system (that is consistent), there exist mathematical ‘truths’ that can’t be proved. It’s the word axiom that is the key, because, in principle, if one extends the axioms one can possibly find a proof.

 

Extending axioms extends mathematics, which is what we’ve done historically since the Ancient Greeks. I referenced Norman Wildberger earlier, and what I believe he’s attempting with his ‘crusade’, is to limit the axioms we’ve adopted, although he doesn’t specifically say that.

 

Someone on Quora recently claimed that we can have ‘contradictory axioms’, and gave Euclidian and subsequent geometries as an example. However, I would argue that non-Euclidean (curved) geometries require new axioms, wherein Euclidean (flat) geometry becomes a special case. As I said earlier, I don’t believe new discoveries prove previous discoveries untrue; they just augment them.

 

But the very employment of axioms, begs a question that no one I listened to addressed: didn’t we humans invent the axioms? And if the axioms are the basis of all the mathematics we know, doesn’t that mean we invented mathematics?

 

Let’s look at some examples. As hard as it is to believe, there was a time when mathematicians were sceptical about negative numbers in the same way that many people today are sceptical about imaginary numbers (i = √-1). If you go back to the days of Plato and his Academy, geometry was held in higher regard than arithmetic, because geometry could demonstrate the ‘existence’, if not the value, of incalculable numbers like π and √2. But negative numbers had no meaning in geometry: what is a negative area or a negative volume?

 

But mathematical ‘inventions’ like negative numbers and imaginary numbers allowed people to solve problems that were hitherto unsolvable, which was the impetus for their conceptual emergence. In both of these cases and the example of non-Euclidean geometry, whole new fields of mathematics opened up for further exploration. But, also, in these specific examples, we were adding to what we already knew. I would contend that the axioms themselves are part of the exploration. If one sees the Platonic world of mathematics as a landscape that only sufficiently intelligent entities can navigate, then axioms are an intrinsic part of the landscape and not human projections.

 

And, in a roundabout way, this brings me back to my introduction concerning the numbers that we discovered as children, whereby we saw a connection between an abstract concept and the physical world. James Franklin, whom I referenced earlier, gave the example of how we measure an area in our backyard to determine if we can fit a shed into the space, thereby arguing the case that mathematics at a fundamental level, and as it is practiced, is dependent on physical parameters. However, what that demonstrates to me is that mathematics determines the limits of what’s physically possible and not the other way round. And this is true whether you’re talking about the origins of the Universe, the life-giving activity of the Sun or the quantum mechanical substrate that underlies our entire existence.

Footnote: Daniel Sutherland (Professor of Philosophy at the University of Illinois, Chicago) adopts the broad category approach that I did, only in more detail. He also points out the 'certainty' of mathematical knowledge that I referenced in the main text. Curiously, he argues that the philosophy of mathematics has influenced the whole of Western philosophy, historically.

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Is gravity a force?

 Believe it or not, this is a question that even physicists don’t seem to agree on. Viktor T Toth, whom I’ve referenced in previous posts and whom I follow on Quora, is almost dismissive of the question. Paraphrasing, he said something like, ‘If you don’t think gravity is a force, just drop a rock onto your foot.’ Ouch! I’d say it’s one of those paradoxes that seem to pop up everywhere in our comprehension of the Universe. 

This question takes me back to my teenage years when I first encountered Newton’s universal theory of gravitation. I struggled to understand how all bodies from the most massive to the smallest could all fall at the same rate in the same gravitational field – it made no sense to me.

 

The equation F = ma is one of the most basic in physics and embodies Newton’s second law in a succinct formula that is universal – it applies everywhere in the Universe. 

 

Leaving gravity aside for the moment, if you apply a uniform force to different masses they will accelerate at different rates. But in gravity, we get a converse relationship. We have different forces dependent on the mass, so we always get the same acceleration. It’s like gravity adjusts its ‘force’ according to the mass it effects. This was the dilemma I couldn’t resolve as a high school physics student.

 

Around the same time, I remember watching a documentary on Einstein (when TV was still in black and white), which was inspiring and thought-provoking in equal measure. One of the more mind-bending ideas I remember was someone explaining how Einstein had changed the concept of gravity as a force to gravity as a curve. This made absolutely no sense to me and it says a lot that I still remember it after more than 5 decades. Of course, I didn’t realise at the time, that these 2 conundrums I was contemplating were related.

 

I’ve recently been reading a book called Emmy Noether’s Wonderful Theorem by Dwight E. Neuenschwander, which is really a university-level text book, not a book for laypeople like me. Emmy Noether famously developed a mathematical formulation to show the relationship between conservation laws and symmetry. Basically, energy is conserved with transformations in time, momentum is conserved with transformations in space and angular momentum is conserved with transformations in rotation. She originally developed this for relativity theory but it equally applies to quantum mechanics. As someone who is not known outside of the physics community, she’s had an enduring and significant input into that field. In her own time, she was not even paid for giving lectures to students, such was the level of prejudice towards women in the sciences in her time (pre WW2).

 

Reading Neuenschwander’s book, I was surprised to learn how much a role the Lagrangian played in her work. Relevant to this topic, Noether was the first to apply the Lagrangian to general relativity, which is actually the easiest way to understand it.

 

To quote from Neuenschwander:

 

The general theory of relativity came along in 1915, and by 1918 the equation of motion of a particle falling in a gravitational field was shown, especially by Emmy Noether and David Hilbert, to be derivable from a variational principle: The world line of a freely falling particle would be that for which the elapsed time between two events was maximised, making the world line a geodesic in spacetime.

 

David Hilbert was Noether’s mentor, and also her greatest champion it has to be said. Hilbert was arguably the greatest living mathematician of his day. The last part of that quote is effectively a description of the Lagrangian, which the author compares to Fermat’s principle. In other words, Fermat’s principle gives the path of least time for light being refracted, and the Lagrangian in a gravitational field gives the geodesic, which is the path of maximum elapsed time for a particle.

 

Notice that in that description of a particle following a geodesic in a gravitational field there is no mention of its mass. This means that its path and its elapsed time is independent of its mass. In fact, we even know it applies to photons, which are, to all intents and purposes, massless.

 

And this revelation finally resolves the conundrum I wrestled with in high school. If you are in free fall, say in a falling lift, or in an orbiting space station; in both cases, you will experience weightlessness.

 

So where is the force? The force is experienced when an object is stopped from following the geodesic, which is the normal everyday experience we all have and don’t even think about. We call it the force of gravity, because gravity is the underlying cause, but the force is actually created by whatever it is that is stopping you from falling, and not the other way round. And, of course, that force is proportional to your mass otherwise you wouldn’t be stationary, relative to whatever’s holding you up. And if you drop a rock on your foot (not recommended) you’ll experience a force that is directly proportional to its mass; no surprise there either.

This is a video, by someone who knows more about this than me, which is even more mind-bending. He argues, quite convincingly, that we are all accelerating by just standing still - on Earth.

Reality, nature, the Universe and everything abhors a contradiction

 People used to say that ‘nature abhors a vacuum’, which is more often than not used metaphorically. Arguably, contradiction avoidance is more fundamental in that you don’t even need a universe for it to be requisite. Many mathematical proofs are premised on the unstated axiom that you can’t have a contradiction - reductio ad absurdum.

However, the study of physics has revealed that nature seems to love paradoxes and the difference between a paradox and a contradiction is often subtle and sometimes inexplicable. So, following that criterion, I believe that reality exists in that sliver of possibility between paradox and contradiction.

 

Clifford A Pickover, who normally writes about mathematics and physics, wrote an entertaining and provocative book, The Paradox of God and the Science of Omniscience. One of the conclusions that I took from that book is that there are rules of logic that even God can’t break. And one of those rules is the rule of contradiction that something can’t ‘be’ and ‘not be’ at the same time. It turns the argument on its head that God created logic. 

 

It’s obvious to anyone who reads this blog that I’m unceasingly fascinated by science and philosophy, and, in particular, where they meet and possibly crossover. So an unerring criterion for me is that you can’t have a contradiction. My recent exposition on the famous twin paradox demonstrates this. It can only be resolved if one accepts that only one twin experiences time dilation. If they both did, you would arrive at a contradiction, both mathematically and conceptually. Specifically, each twin would perceive the other one as younger, which is impossible. 

 

The opposite to contradiction is consistency, and if you look at the mathematical analysis of the twin paradox, you’ll see it’s consistent throughout for both twins.

 

To give another example, physicists tell us that time does not flow (as Paul Davies points out in this presentation, 48.30min mark), yet we all experience time ‘flowing’. Davies and I agree that the sensation we have of time ‘flowing’ is a psychological experience; we disagree on how or why that happens. 

 

Time is a dimension determined by the speed of light. Everything is separated, not only spatially, but also in time, because it takes a finite time for light to travel between events separated in space. This leads to the concept of spacetime, which is invariant for different observers, while space and time (independently) can be ‘measured’ to be different for different observers. Spacetime is effectively an extension of Pythagoras’s theorem into 4 dimensions, only it involves negatives.

Δs² = c²Δt² - Δx² - Δy² - Δz²

So Δs is the ‘interval’ that remains invariant, and cΔt is the time dimension converted into a spatial dimension, otherwise the equation wouldn’t work. But note that it’s c (the constant speed of light) that makes time a dimension, and it’s one of the dimensions of the Universe as a whole. Einstein’s equations work for the entire Universe, which is his greatest legacy.

 

I’ve been reading Brian Greene’s The Fabric of the Cosmos (2004), which I came across while browsing in a bookshop and bought it on impulse – I’m glad I did. It’s a 500 page book, covering all the relevant topics on cosmology, so it’s very ambitious, but also very readable.

 

Theories of gravity started with Newton, and he came up with a thought experiment that was still unresolved when Einstein revolutionised his theory. Greene discusses it in some detail. If you take a bucket of water and hang it from a rope, then turn the bucket many times so that the rope is twisted. When you release the rope the bucket spins and the water surface becomes concave in the bucket due to the centrifugal force. The point is, what is the reference frame that the bucket is spinning to? Is it the Universe as a whole? Newton would have argued that it was spinning relative to absolute space.

 

Now imagine that you could do this experiment in space; only, instead of a bucket of water, you have 2 rocks tied together. You could do it on the space station. As you spin them, you’d expect the rope or cord between them to tension. So again, what are they spinning in reference to? Greene gives a detailed historical account because it involves Mach’s principle. But in the end, when Einstein applied his theories of relativity, he came to the conclusion that they spin relative to spacetime. While we don’t have absolute space and absolute time, we have absolute spacetime, according to Einstein.

 

Why have I taken so much trouble to describe this? Because there is a frame of reference that is used to determine what direction and what speed our solar system travels in the context of the overall Universe. And it’s the Cosmic Microwave Background Radiation. Paul Davies described this in his book, About Time (1995). By measuring the difference in the Doppler effect (for CMBR) in different regions of the sky, we can deduce we are travelling in the direction of the Pisces constellation. Greene also points this out in his book.

 

All physicists that I’ve read, or listened to, argue that ‘now’ is totally dependent on the observer. They will tell you that if you move backwards and forwards here on Earth, then the time changes on some far off constellation will be in the order of hundreds of years because of the change in angle of the time slice across that part of the Universe. Greene himself explains this in a video, as well as in his book. But there is an ‘age’ for the Universe, so you have an implicit contradiction. Greene is the only person I’ve read who actually attempts to address this, though not very satisfactorily (for me).

 

But one doesn’t need to look at far off galaxies, one can look at Einstein’s original thought experiment involving 2 observers: one on a train and one on a platform. The reason physicists argue that there is no objective now is because simultaneity is different according to different observers, and Einstein uses a train as a thought experiment to demonstrate this.

 

If you have a light source in the centre of a moving carriage then a person in the carriage will observe that it gets to both ends at the same time. But a person on the platform will see that the back of the train will receive the light before the front of the train. However, the light source is moving relative to the observer on the platform, so they will see a Doppler shift in the light showing that it is moving relative to them. I contend that only the observer in the same frame of reference as the light source sees the true simultaneity. In other words, I argue that you can have 'true simultaneity' in the same way as you can have ‘true time’. Also, what many people don’t realise, that different observers not only see simultaneity happening at different times but different locations, as this video demonstrates.

 

I’m a subscriber to Quora, mainly because I get to read posts by lots of people in various fields, most of whom are more knowledgeable than me. In fact, I claim my only credential is that I read a lot of books by people much smarter than me. One of the regular contributors to Quora is Viktor T Toth, whom I’ve referenced before, and who calls himself a ‘part-time physicist’. Toth knows a lot about cosmology, QFT (quantum field theory) and black holes. He occasionally shows his considerable mathematical abilities in dealing with a question, but most of the time keeps them in reserve. What I like about Toth, is not just his considerable knowledge, but his no-nonsense approach. He doesn’t pretend or bluff; he has no problem admitting what he doesn’t know and is very respectful to his peers, while not tolerating fools.

 

Toth points out that while 2 observers can experience different durations in time (like the twins in the twin paradox) they agree on the time when they meet again. In other words, there is a time reference (like a space reference) that’s independent of the path they took to get there. As I pointed out in my post explaining relativity based on waves, it’s not only the time duration that 2 observers disagree on, but also the space duration. If someone was able to travel so fast that they could cross the galaxy in years instead of thousands of years, then they would also traverse a much shorter distance (according to them). In other words, not only does time shrink, but so does distance. We don’t tend to think that space can be just as rubbery as time.

 

And this brings me to a point that Toth and Greene seem to disagree on. Toth is adamant that space is not an entity but just the ‘distance’ between physical objects. Regarding the expanding universe, he says ‘space’ does not ‘stretch’ but it’s just that the ‘distance’ between ‘objects’ increases. Greene would probably disagree. Greene argues that wavelengths of light lengthen as space ‘stretches’.

 

What do I think? I tend to side with Greene. I think space is an entity because it has dimensions. John Barrow, in his book, The Constants of Nature, gives an excellent account of how a universe that didn’t exist in 3 dimensions would be virtually unworkable. In particular, the inverse square law for gravity, that keeps planets in stable orbits for hundreds of millions of years, would not work in any other dimensional universe.

 

But also, space can be curved by gravity, or more specifically, spacetime, as Toth readily acknowledges. I’ll return to Toth’s specific commentary on gravity and black holes later.

 

I’ve already mentioned that Greene discussed the age of the Universe. I recently did an online course provided by New Scientist on The Cosmos, and one of the lecturers was Chris Impey, Distinguished Professor, Department of Astronomy, University of Arizona. He made the point that the Universe has an ‘edge in time’, but not an edge in space. Greene expanded on this, by pointing out that everywhere in the Universe all clocks are ‘in synchronicity’, which contradicts the notion that there is no universal now. I’ll quote Greene directly on this, because it’s an important point. According to Greene (but not only Greene) whether an observer is moving away from a distant stellar object, or towards it, determines whether they would see into that object’s distant past or distant future. Mind you, because they are so far away, the object’s future is still in our past.

 

Each angled slice intersects the Universe in a range of different epochs and so the slices are far from uniform. This significantly complicates the description of cosmic history, which is why physicists and astronomers generally don’t contemplate such perspectives. Instead, they usually consider only the perspective of observers moving solely with the cosmic flow...

 

I don’t know the mathematics behind this, but I can think of an analogy that we all observe every day. You know when you walk along a street with the Sun low in the sky, so it seems to be moving with you. It will disappear behind a building then appear on the other side. And of course, another observer in another town will see the Sun in a completely different location with respect to their horizon. Does this mean that the Sun moved thousands of miles while you were walking along? No, of course not. It’s all to do with the angle of projection. In other words, the movement in the sky is an illusion, and we all know this because it happens all the time and it happens in sync with our own movements. I remember once travelling in a car with a passenger and we could see a plane low in the sky through the windscreen. My passenger commented that the plane was travelling really fast, and I pointed out that if we stopped, we’d find that the plane would suddenly slow down to a speed more commensurate with our expectations.

 

I think the phenomenon that Greene describes is a similar illusion, only we conjure it up mathematically. It makes sense to ignore it, as astronomers do (as he points out) because we don’t really expect it happens in actual fact.

 

Back in 2016, I wrote a post on a lecture by Allan Adams as part of MIT Open Courseware (8.04, Spring 2013) titled Lecture 6: Time Evolution and the Schrodinger Equation. This was a lecture for physics students, not for a lay audience. I found this very edifying, not least because it became obvious to me, from Adams’ exposition, that you could have a wave function with superposition as described by Schrodinger’s equation or you could have an observed particle (like an electron) but you couldn’t have both. Then I came across the famous quote by William Lawrence Bragg:

Everything that has already happened is particles, everything in the future is waves. The advancing sieve of time coagulates waves into particles at the moment ‘now’.

 

In that post on Adams’ lecture, I said how people (like Roger Penrose, among others) explained that ‘time’ in the famous time dependent Schrodinger equation exists outside the hypothetical Hilbert space where the wave function hypothetically exists. And it occurred to me that maybe that’s because the wave function exists in the future. And then I came across Freeman Dyson’s lecture and his unorthodox claim:

 

... the “role of the observer” in quantum mechanics is solely to make the distinction between past and future...

What really happens is that the quantum-mechanical description of an event ceases to be meaningful as the observer changes the point of reference from before the event to after it. 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 inherent contradictions in attempting to incorporate classical physics into quantum mechanics (QM) disappear, if one is describing all the possible future paths of an event while the other describes what actually happened.

 

Viktor Toth, whom I’ve already mentioned, once made the interesting contention that the wave function is just a ‘mathematical construct’ (his words) and he’s not alone. He also argued that the ‘decoherence’ of the wave function is never observed, which infers that it’s already happened. Toth knows a great deal about QFC (and I don’t) but, if I understand him correctly, the field exists all the time and everywhere in spacetime.

 

Another contributor on Quora, whom I follow, is Mark John Fernee (PhD in physics, University of Queensland), who obviously knows a great deal more than me. He had this to say about ‘wave function collapse’ (or decoherence) which corroborates Toth.

 

The problem is that there is no means to detect the wavefunction, and consequently no way to detect a collapse. The collapse hypothesis is just an inference that can't be experimentally tested.

 

And, in a comment, he made this point:

 

In quantum mechanics, the measurement hypothesis, which includes the collapse of the wave function, is an irreversible process. As we perceive the world through measurements, time will naturally seem irreversible to us.

 

And I’ve made this exact same point, that we also get the ‘irreversibility’ or asymmetry between past and future, from QM physics becoming classical physics.

 

Dyson is also contentious when it comes to gravity and QM, arguing that we don’t need to combine them together and that, even if the graviton existed, it’s impossible to detect. Most physicists argue that we need a quantum gravity – that it’s the ‘missing link’ in a TOE (Theory of Everything).

 

It’s occurred to me that maybe the mathematics is telling us something when it appears obstructive to marrying general relativity with QM. Maybe they don’t go together, as Dyson intimated.

 

Again, I think Toth gives the best reasoning on this, as I elaborated on in another post.

 

We can do quantum field theory just fine on the curved spacetime background of general relativity.

Then he adds this caveat:

What we have so far been unable to do in a convincing manner is turn gravity itself into a quantum field theory.

 

Toth explains how and why they are ‘incompatible’, though he loathes that term, because he doesn’t think they’re incompatible at all.

 

Basically, Einstein’s field equations are geometrical, whereby one side of the equation gives the curvature of spacetime as a consequence of the energy, expressed on the other side of the equation. The energy, of course, can be expressed in QM, but the geometry can’t. 

 

Someone else on Quora, Terry Bollinger (retired Chief Scientist), explains this better than me:

 

It all goes back to that earlier point that GR is a purely geometric theory, which in turn means that the gravity force that it describes is also specified purely in terms of geometry. There are no particles in gravity itself, and in fact nothing even slightly quantum. Instead, you assume the existence of a smooth fabric called spacetime, and then start bending it. From that bending emerges the force we know as gravity.

 

Bollinger wrote a lengthy polemic on this, but I will leave you with his conclusion, because it is remarkably similar to Toth’s.

 

Even if you finally figure out a clever way to define a gravity-like quantum force that allows objects in spacetime to attract each other, what are those force particles traveling across?

 

He provides his own answer:

 

Underneath the quantum version, since like all of the other forces in the Standard Model this swarm-like quantum version of a gravity must ride on top of spacetime. (Emphasis in the original)

 

In other words, Bollinger claims you’ll have a redundancy: a quantum field gravity on top of a spacetime gravity. Spacetime provides the ‘background’ to QFT that Toth described, which doesn’t have to be quantum.

 

So what about quantum gravity that is apparently needed to explain black holes?

 

There is an inherent contradiction in relation to a black hole, and I’m not talking about the ‘information paradox’. There is a debate about whether information (meaning quantum information) gets lost in a black hole, which contradicts the conservation of quantum information. I’m not going to get into that as I don’t know enough about it.

 

There is a more fundamental issue: according to Toth (but not only Toth) the event horizon of a black hole is always in an observer’s future. Yet for someone at the event horizon, they could cross over it without even knowing they had done so, especially if it was a really big black hole and the tidal effects at the event horizon weren’t strong enough to spaghettify them. Of course, no one really knows this, because no one has ever been anywhere near a black hole event horizon.

 

The point is that, at the event horizon of a black hole, time stops according to an external observer, which is why, theoretically it’s always in their future. Toth makes this point many times. So, basically, for someone watching something fall into a black hole, it becomes frozen in time (at the event horizon). In fact, the Russian term for a black hole is ‘frozen star’.

 

What we can say is that light from any object gets red-shifted so much that it disappears even before it reaches the event horizon. But what about an observer at or near the event horizon looking back out at the Universe. Now, I don’t see any difference in this scenario to the twin paradox, only it’s in extremis. The observer at the event horizon, or just outside it, will see the whole universe pass by in their lifetime. Because, if they could come back and meet up with their twin, their twin would be a hologram frozen in time, thousands of years old. Now, Toth makes the point, that as far as an observer at or near the event horizon, the speed of light is still constant for them, so how can that be?

 

There is another horizon in the Universe, which is the theoretical and absolute practical limit that we can see. Because the Universe is expanding, there is a part of the Universe that is expanding faster than light, relative to us. Now, you will say, how is that possible? It’s possible because space can travel faster than light. Now Toth will confirm this, even though he claims space is not an entity. Note that some other observer in a completely different location, would see a different horizon, in the same way that sailors in different locations in the same ocean see different horizons.

 

So, a hypothetical observer, at the horizon of the Universe (with respect to us) would still see the speed of light as c relative to their spacetime. Likewise, an observer at the event horizon of a black hole also sees light as c relative to their spacetime.

 

Now most black holes, we assume, are spinning black holes and they drag an accretion disk around with them. They also drag space around with them. Is it possible then that the black hole drags space along with an observer across the event horizon into the black hole? I don’t know, but it would resolve the paradox. According to someone on Quora, Leonard Susskind argues that nothing ever crosses the event horizon, which is how he resolves the quantum information paradox.

 

This is a very lengthy post, even by my standards, but I need to say something quickly about entanglement. There appears to be a contradiction between relativity and entanglement, but not in practical terms. If there is a universal 'now', implicit in the Universe having an ‘edge in time’ (but not in space) and if QM describes the future, then entanglement is not a mystery, because it’s a correlation between events separated in space, but not in time. 

 

Entanglement involves a ‘decoherence’ in the wave function that predicts the state of a decoherence in the particle it’s entangled with, because they share the same wave function. Schrodinger understood this better than anyone else, because he realised that entanglement was an intrinsic consequence of the wave function. He famously said that entanglement was the defining characteristic of quantum mechanics.

 

But there is no conflict with relativity because the entangled particles, whatever they are, can’t be separated at any greater rate than the speed of light. However, when the correlation occurs, it appears to happen instantaneously. But this is no different to a photon always being in the future of whatever it interacts with, even if it crosses the observable universe. However, for someone who detects such a photon, they instantaneously see something in the distant past as if there has been a backward-in-time connection to its source.

 

There is no reason to believe that anything I’ve said is true and correct. I’ve tried to follow a simple dictum that nature abhors a contradiction and apply it to what I know about the Universe, while acknowledging there are lots of people who know a great deal more than me, who probably disagree.

 

I see myself as an observer on the boundary line of the history of ideas. I try to make sense of the Universe by reading and listening to people much cleverer than me, including people I have philosophical differences with.

Addendum 1: I referenced Paul Davies 1995 book, About Time; Einstein's Unfinished Revolution. I mentioned that Earth is travelling relative to the CMBR towards Pisces (at 350 km/s), and according to Davies:

This is about 0.1 percent of the speed of light, and the time-dilation factor is only about one part in a million. Thus, to an excellent approximation, Earth's historical time coincides with cosmic time, so we can recount the history of the universe contemporaneously with the history of the Earth, in spite of the relativity of time. Similar hypothetical clocks could be located everywhere in the universe, in each case in a reference frame where the cosmic background heat radiation looks uniform... we can imagine the clocks out there, and legions of sentient beings dutifully inspecting them. This set of imaginary observers will agree on a common time scale and a common set of dates for major events in the universe, even though they are moving relative to each other as a result of the general expansion of the universe. They could cross-check dates and events by sending each other data by radio; everything would be consistent. So cosmic time as measured by this special set of observers constitutes a type of universal time... It is the existence of this pervasive time scale that enables cosmologists to put dates to events in cosmic history - indeed, to talk meaningfully at all about "the universe" as a single system. (my emphasis)

Addendum 2: This is a PBS video, which gives the conventional physics view on time. I don't know who the presenter is, but it would be fair to say he knows more about this topic than me. He effectively explains Einstein's 'block universe' and why 'now' is considered totally subjective. Remember Einstein's famous words in a letter to the mother of a friend who had died:

We physicists know that the past, present and future is only a stubbornly persistent illusion.

This was a consequence of simultaneity being different for different observers, as I discussed in the main text, and is described in the video. It's important to point out that this does not undermine causality, so it refers to events that are not causally related. The video presenter goes on to point out that different observers will see different pasts and different futures on worlds far far away, dependent on their motion on this world. This infers that all events are predetermined, which is what Einstein believed, and explains why so many physicists claim that the Universe is deterministic. But it contradicts the view, among cosmologists, that the Universe has an 'edge in time but not in space'.

It's certainly worth watching the video. Curiously, his logic leads him to the conclusion that we live in a quantum multiverse (the many worlds interpretation of QM). I agree with him that different observers in different parts of the Universe must have different views of 'Now'. That's just a logical consequence of the finite speed of light. Motion then distorts that further, as he demonstrates. My view is that what we perceive is not necessarily what actually 'is'. If one looks at the clock of a moving observer their time is dilated compared to ours, and likewise they see our clocks showing time dilation compared to them. But logic tells us that they both can't be right. The twin paradox is resolved only if one acknowledges that time dilation is an illusion for one observer but not the other. And that's because one of the twin travels relative to an absolute spacetime if not an absolute space or an absolute time.

Back to the video, my contention is that one observer can't see another observer's future, even though we can see another observer's 'present' in our 'past'; I don't find that contentious at all.

The closest I’ve ever seen to someone explaining my philosophy

 I came across this 8min video of Paul Davies from 5 years ago, where I was surprised to find that he and I had very similar ideas regarding the ‘Purpose’ of the Universe. In more recent videos, he has lighter hair and has lost his moustache, which was a characteristic of his for as long as I’ve followed him.

 

Now, one might think that I shouldn’t be surprised, as I’ve been heavily influenced by Davies over many years (decades even) and read many of his books. But I thought he was a Deist, and maybe he was, because not halfway through he admits he had recently changed his views.

 

But what makes me consider that this video probably comes closest to expressing my own philosophy is when he says that meaning or purpose has evolved and that it’s directly related to the fact that the Universe created the means to understand itself. Both these points I’ve been making for years. In his own words, “We unravel the plot”.

Or to quote John Wheeler (whom Davies admired): “The universe gives birth to consciousness, and consciousness gives meaning to the universe.”

P.S. This is also worth watching: his philosophy on mathematics; to which I would concur. His metaphor of a 'warehouse' is unusual, yet very descriptive and germane in my view.

The Twin paradox, from both sides now (with apologies to Joni)

 I will give an exposition on the twin paradox, using an example I read in a book about 4 decades ago, so I’m relating this from memory.

Imagine that one of the twins goes to visit an extra-terrestrial world 20 light years away in a spaceship that can travel at 4/5 the speed of light. The figures are chosen because they are easy to work with and we assume that acceleration and stopping are instantaneous. We also assume that the twin starts the return journey as soon as they arrive at their destination.

 

From the perspective of the twin on Earth, the trip one-way takes 25 years because the duration is T = s/v, where s = 20 (light years) and v = 4/5c. 

So 5/4 x 20 = 25.

 

From the perspective of the twin on the spaceship, their time is determined by the Lorentz transformation (γ).

 

γ = 1/√(1 – v²/c²)

 

Note v²/c² = (4/5)² = 16/25

So √(1 – v²/c²) = √(9/25) = 3/5

 

Now true time for the space ship (τ) is given by τ = T/γ

So for the spaceship twin, the duration of the trip is 3/5 x 25 = 15

So the Earth twin has aged 25 years and the spaceship twin has aged 15 years.

 

But there is a relativistic Doppler effect, which can be worked out by considering what each twin sees when the spaceship arrives at its destination. 

 

Note that light, or any other signal, takes 20 years to come back from the destination. So the Earth twin will see the space ship arrive 25 + 20 = 45 years after it departed. But they will see that their twin is only 15 years older than when they left. So, from the Earth twin’s perspective, the Doppler effect is a factor of 3. (3 x 15 = 45). So the Doppler effect slowed time down by 3. Note: 45 years has passed but they see their twin has only aged 1/3 of that time.

 

What about the spaceship twin’s perspective? They took 15 years to get there, but the Doppler effect is a factor of 3 for them as well. They’ve been receiving signals from Earth ever since they left so they will see their twin only 5 years older because 15/3 = 5, which is consistent with what their twin saw. In other words, their Earth twin has aged 5 years in 15, or 1/3 of their travel time. 

 

If spaceship twin was to wait another 20 years for the signal to arrive then it would show Earth twin had aged 5 + 20 = 25 years at the time of their arrival. But, of course, they don’t wait, they immediately return home. Note that the twins would actually agree on each other’s age if they allowed for the time it takes light to arrive to their respective locations.

 

So what happens on the return trip? The Lorentz transformation is the same for the spaceship twin on the return trip, so they only age another 15 years, but according to the Earth twin the trip would take another 25 years, so they would have aged 50 years compared to the 30 years of their twin.

 

But what about the Doppler effect? Well, it’s still a factor of 3, only now it works in reverse, speeding time up. For the spaceship twin, their 15 years of observing their Earth twin is factored by 3 and 15 x 3 = 45. And 5 + 45 = 50, which is how much older their twin is when they arrive home.

 

For the Earth twin, their spaceship twin’s round trip is 50 years, so the return trip appears to only take 5 years. And allowing for the same Doppler effect, 3 x 5 = 15.

When the Earth twin adds 15 to the 15 years they saw after 45 years, they deduce the age of their spaceship twin is 30 years more (against their own 50). So both twins are in agreement.

 

Now, the elephant in the room is why do we only apply the Lorentz transformation to the spaceship twin? The usual answer to this question is that the spaceship twin had to accelerate and turn around to come back, so it’s obvious they did the travelling.

 

But I have another answer. The spaceship twin leaves the surface of Earth and even leaves the solar system. It’s obvious that the spaceship didn’t remain stationary while the solar system travelled through the cosmos at 4/5 the speed of light. There is an asymmetry to the scenario which is ultimately governed by the gravitational field created by everything, and dominated by the solar system in this particular case. In other words, the Lorentz transformation only applies to the spaceship twin, even when they only travel one way.

Relativity makes sense if everything is wavelike

 When I first encountered relativity theory, I took an unusual approach. The point is that c can always be constant while the wavelength (λ) and frequency (f ) can change accordingly, because c = λ x f. This is a direct consequence of v = s/t (where v is velocity, s distance and t time). We all know that velocity (or speed) is just distance divided by time. And λ represents distance while f represents 1/t. 

So, here’s the thing: it occurred to me that while wavelength and frequency would change according to the observer’s frame of reference (meaning relative velocity to the source), the number of waves over a specific distance would be the same for both, even though it’s impossible to measure the number of waves. And a logical consequence of the change in wavelength and frequency is that the observers would ‘measure’ different distances and different periods of time.

 

One of the first confirmations of relativity theory was to measure the half-lives of cosmic rays travelling through the Earth’s atmosphere to reach a detector at ground level. Measurements showed that more particles arrived than predicted by their half-life when stationary. However, allowing for relativistic effects (as the particles travelled at high fractional lightspeeds), the number of particles detected corresponded to time dilation (half-life longer, so more particles arrived). This means from the perspective of the observers on the ground, if the particles were waves, then the frequency slowed, which equates to time dilation - clocks slowing down. It also means that the wavelength was longer so the distance they travelled was further. 

 

If the particles travelled slower (or faster), then wavelength and frequency would change accordingly, but the number of waves would be the same. Of course, no one takes this approach - why would you calculate the Lorentz transformation on wavelength and frequency and multiply by the number of waves, when you could just do the same calculation on the overall distance and time.

 

Of course, when it comes to signals of communication, they all travel at c, and changes in frequency and wavelength also occur as a consequence of the Doppler effect. This can create confusion in that some people naively believe that relativity can be explained by the Doppler effect. However, the Doppler effect changes according to the direction something or someone is travelling while relativistic effects are independent of direction. If you come across a decent mathematical analysis of the famous ‘twin paradox’, you’ll find it allows for both the Doppler effect and relativistic effects, so don’t get them confused.

 

Back to the cosmic particles: from their inertial perspective, they are stationary and the Earth with its atmosphere is travelling at high fractional lightspeed relative to them. So the frequency of their internal clock would be the same as if they were stationary, which is higher than what the observers on the ground would have deduced. Using the wave analogy, higher frequency means shorter wavelength, so the particles would ‘experience’ the distance to the Earth’s surface as shorter, but again, the number of waves would be the same for all observers.

 

I’m not saying we should think of all objects as behaving like waves - despite the allusion in the title - but Einstein always referred to clocks and rulers. If one thinks of these clocks and rulers in terms of frequencies and wavelengths, then the mathematical analogy of a constant number of waves is an extension of that. It’s really just a mathematical trick, which allows one to visualise what’s happening.

What is scientism?

 I’m currently reading a book (almost finished, actually) by Hugh Mackay, The Inner Self; The joy of discovering who we really are. Mackay is a psychologist but he writes very philosophically, and the only other book of his I’ve read is Right & Wrong: How to decide for yourself, which I’d recommend. In particular, I liked his chapter titled, The most damaging lies are the ones we tell ourselves.

You may wonder what this has to do with the topic, but I need to provide context. In The Inner Self, Mackay describes 20 ‘hiding places’ (his term) where we hide from our ‘true selves’. It’s all about living a more ‘authentic’ life, which I’d endorse. To give a flavour, hiding places include work, perfectionism, projection, narcissism, victimhood – you get the picture. Another term one could use is ‘obsession’. I recently watched a panel of elite athletes (all Australian) answering a series of public-sourced questions (for an ABC series called, You Can’t Ask That), and one of the take-home messages was that to excel in any field, internationally, you have to be obsessed to the point of self-sacrifice. But this also applies to other fields, like performing arts and scientific research. I’d even say that writing a novel requires an element of obsession. So, obsession is often a necessity for success.

 

With that caveat, I found Mackay’s book very insightful and thought-provoking – It will make you examine yourself, which is no bad thing. I didn’t find any of it terribly contentious until I reached his third-last ‘hiding place’, which was Religion and Science. The fact that he would put them together, in the same category, immediately evoked my dissent. In fact, his elaboration on the topic bordered on relativism, which has led me to write this post in response.

 

Many years ago (over 2 decades) when I studied philosophy, I took a unit that literally taught relativism, though that term was never used. I’m talking epistemological relativism as opposed to moral relativism. It’s effectively the view that no particular discipline or system of knowledge has a privileged or superior position. Yes, that viewpoint can be found in academia (at least, back then).

 

Mackay’s chapter on the topic has the same flavour, which allows him to include ‘scientism’ as effectively a religion. He starts off by pointing out that science has been used to commit atrocities the same as religion has, which is true. Science, at base, is all about knowledge, and knowledge can be used for good or evil, as we all know. But the ethics involved has more to do with politicians, lawmakers and board appointees. There are, of course, ethical arguments about GM foods, vaccinations and the use of animals in research. Regarding the last one, I couldn’t personally do research involving the harming of animals, not that I’ve ever done any form of research.

 

But this isn’t my main contention. He makes an offhand reference at one point about the ‘incompatibility’ of science and religion, as if it’s a pejorative remark that reflects an unjustified prejudice on the part of someone who’d make that comment. Well, to the extent that many religions are mythologically based, including religious texts (like the Bible), I’d say the prejudice is justified. It’s what the evolution versus creation debate is all about in the wealthiest and most technologically advanced nation in the world.

 

I’ve long argued that science is neutral on whether God exists or not. So let me talk about God before I talk about science. I contend that there are 2 different ideas of God that are commonly conflated. One is God as demiurge, and on that I’m an atheist. By which I mean, I don’t believe there is an anthropomorphic super-being who created a universe just for us. So I’m not even agnostic on that, though I’m agnostic about an after-life, because we simply don’t know.

 

The other idea of God is a personal subjective experience which is individually unique, and most likely a projection of an ‘ideal self’, yet feels external. This is very common, across cultures, and on this, I’m a theist. The best example I can think of is the famous mathematician, Srinivasa Ramanujan, who believed that all his mathematical insights and discoveries came directly from the Hindu Goddess, Namagiri Thayar. Ramanujan (pronounced rama-nu-jan) was both a genius and a mystic. His famous ‘notebooks’ are still providing fertile material 100 years later. He traversed cultures in a way that probably wouldn’t happen today.

 

Speaking of mathematics, I wrote a post called Mathematics as religion, based on John Barrow’s book, Pi in the Sky. According to Marcus du Sautoy, Barrow is Christian, though you wouldn’t know it from his popular science books. Einstein claimed he was religious, ‘but not in the conventional sense’. Schrodinger studied the Hindu Upanishads, which he revealed in his short tome, Mind and Matter (compiled with What is Life?).

 

Many scientists have religious beliefs, but the pursuit of science is atheistic by necessity. Once you bring God into science as an explanation for something, you are effectively saying, we can’t explain this and we’ve come to the end of science. It’s commonly called the God-of-the-gaps, but I call it the God of ignorance, because that’s exactly what it represents.

 

I have 2 equations tattooed on my arms, which I describe in detail elsewhere, but they effectively encapsulate my 3 worlds philosophy: the physical, the mental and the mathematical. Mackay doesn’t talk about mathematics specifically, which is not surprising, but it has a special place in epistemology. He does compare science to religion in that scientific theories incorporate ‘beliefs', and religious beliefs are 'the religious equivalent of theories'. However, you can’t compare scientific beliefs with religious faith, because one is contingent on future discoveries and the other is dogma. All scientists worthy of the name know how ignorant we are, but the same can’t be said for religious fundamentalists.

 

However, he's right that scientific theories are regularly superseded, though not in the way he infers. All scientific theories have epistemological limits, and new theories, like quantum mechanics and relativity (as examples), extend old theories, like Newtonian mechanics, into new fields without proving them wrong in the fields they already described. And that’s a major difference to just superseding them outright.

 

But mathematics is different. As Freeman Dyson once pointed out, a mathematical theorem is true for all time. New mathematical discoveries don’t prove old mathematical discoveries untrue. Mathematics has a special place in our system of knowledge.

 

So what is scientism? It’s a pejorative term that trivialises and diminishes science as an epistemological success story.