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 contends that all the non-trivial zeros of the Riemann Zeta function lie on a line in the complex plane which is 1/2 + ib. In other words, all the (nontrivial) zeros have Real part 1/2. The thing is that we already know there are an infinite number of them, we just don’t know if there are any that break that rule. The curious thing about infinities is that we are relatively comfortable with them, even though we can’t relate to them in the physical world, and they can never be computed. As I said in my opening paragraph, it’s what separates mathematics from reality.

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

 

Addendum: I made a very simple yet unforgivable mistake (since corrected), whereby I said the Zeta zeros in Riemann's Hypothesis were of the form a + 1/2ib, when it's the other way around: 1/2 + ib. So apologies.

How can I know when I am wrong?

 Simple answer: I can’t. But this goes to the heart of a dilemma that seems to plague the modern world. It’s even been given a name: the post-truth world.  

I’ve just read a book, The Psychology of Stupidity; explained by some of the world’s smartest people, which is a collection of essays by philosophers, psychologists and writers, edited by Jean-Francois Marmion. It was originally French, so translated into English; therefore, most of the contributors are French, but some are American. 

 

I grew up constantly being reminded of how stupid I was, so, logically, I withdrew into an inner world, often fuelled by comic-book fiction. I also took refuge in books, which turned me into a know-it-all; a habit I’ve continued to this day.

 

Philosophy is supposed to be about critical thinking, and I’ve argued elsewhere that critical analysis is what separates philosophy from dogma, but accusing people of not thinking critically does not make them wiser. You can’t convince someone that you’re right and they’re wrong: the very best you can do is make them think outside their own box. And, be aware, that that’s exactly what they’re simultaneously trying to do to you.

 

Where to start? I’m going to start with personal experience – specifically, preparing arguments (called evidence) for lawyers in contractual engineering disputes, in which I’ve had more than a little experience. Basically, I’ve either prepared a claim or defended a claim by analysing data in the form of records – diaries, minutes, photographs – and reached a conclusion that had a trail of logic and evidence to substantiate it. But here’s the thing: I always took the attitude that I’d come up with the same conclusion no matter which side I was on.

 

You’re not supposed to do that, but it has advantages. The client, whom I’m representing, knows I won’t bullshit them and I won’t prepare a case that I know is flawed. And, in some cases, I’ve even won the respect of the opposing side. But you probably won’t be surprised to learn how much pressure you can be put under to present a case based on falsehoods. In the end, it will bite you.

 

The other aspect to all this is that people can get very emotional, and when they get emotional they get irrational. Writing is an art I do well, and when it comes to preparing evidence, my prose is very dispassionate, laying out an argument based on dated documents; better still, if the documents belong to the opposition.

 

But this is doing analysis on mutually recognised data, even if different sides come to different conclusions. And in a legal hearing or mediation, it’s the documentation that wins the argument, not emotive rhetoric. Most debates these days take place on social media platforms where people on opposing sides have their own sources and their own facts and we both accuse each other of being brainwashed. 

 

And this leads me to the first lesson I’ve learned about the post-truth world. In an ingroup-outgroup environment – like politics – even the most intelligent people can become highly irrational. We see everyone on one side as being righteous and worthy of respect, while everyone on the other side is untrustworthy and deceitful. Many people know about the infamous Robbers Cave experiment in 1954, where 2 groups of teenage boys were manipulated into an ingroup-outgroup situation where tensions quickly escalated, though not violently. I’ve observed this in contractual situations many times over.

 

One of my own personal philosophical principles is that beliefs should be dependent on what you know and not the other way round. It seems to me that we do the opposite: we form a belief and then actively look for evidence that turns that belief into knowledge. And, in the current internet age, it’s possible to find evidence for any belief at all, like the Earth being flat.

 

And this has led to a world of alternate universes, where the exact opposite histories are being played out. The best known example is climate change, but there are others. Most recently, we’ve had a disputed presidential election in the USA and the efficaciousness of vaccines in combatting the coronavirus (SARS-Cov-2 or COVD-19). What all these have in common is that each side believes the other side has been duped.

 

You might think that something else these 3 specific examples have in common is left-wing, right-wing politics. But I’ve learned that’s not always the case. One thing I do believe they have in common is open disagreement between purported experts in combination with alleged conspiracy theories. It so happens that I’ve worked with technical experts for most of my working life, plus I read a lot of books and articles by people in scientific disciplines. 

 

I’m well aware that there are a number of people who have expertise that I don’t have and I admit to getting more than a little annoyed with politicians who criticise or dismiss people who obviously have much more expertise than they have in specific fields, like climatology or epidemiology. One only has to look to the US, where the previous POTUS, Donald Trump, was at the centre of all of these issues, where everything he disagreed with was called a ‘hoax’, and who was a serial promoter of conspiracy theories, including election fraud. Trump is responsible for one of those alternative universes where President Elect, Joe Biden, stole the election from him, even though there is ample testimony that Trump tried to steal the election from Biden.

 

So, in the end, it comes down to who do you trust. And you probably trust someone who aligns with your ideological position or who reinforces your beliefs. Of course, I also have political views and my own array of beliefs. So how do I navigate my way?

 

Firstly, I have a healthy scepticism about conspiracy theories, because they require a level of global collaboration that’s hard to maintain in the manner they are reported. They often read or sound like movie scripts, with politicians being blackmailed or having their lives threatened and health professionals involved in a global conspiracy to help an already highly successful leader in the corporate world take control of all of our lives. This came from a so-called ‘whistleblower’, previously associated with WHO.

 

The more emotive and sensationalist a point of view, the more traction it has. Media outlets have always known this, and now it’s exploited on social media, where rules about accountability and credibility are a lot less rigorous.

 

Secondly, there are certain trigger words that warn me that someone is talking bullshit. Like calling vaccines a ‘bio-weapon’ or that it’s the ‘death-jab’ (from different sources). However, I trust people who have a long history of credibility in their field; who have made it their life’s work, in fact. But we live in a world where they can be ridiculed by politicians, whom we are supposed to respect and follow.

 

At the end of the day, I go back to the same criteria I used in preparing arguments involved in contractual disputes, which is evidence. We’ve been living with COVID for 2 years now and it is easy to find statistical data tracking the disease in a variety of countries and the effect the vaccines have had. Of course, the conspiracy theorists will tell you that the data is fabricated. The same goes for evidence involving climate change. There was a famous encounter between physicist and television presenter, Brian Cox, and a little known Australian politician who claimed that the graphs Cox presented, produced by NASA, had been corrupted.

 

But, in both of these cases, the proof is in the eating of the pudding. I live in a country where we followed the medical advice, underwent lockdowns and got vaccinated, and we’re now effectively living with the virus. When I look overseas, at countries like America, it was a disaster overseen by an incompetent President, who advocated all sorts of ‘crank cures’, the most notorious being bleach, not to mention UV light. At one point, the US accounted for more than 20% of the world’s recorded deaths.

 

And it’s the same with climate change where, again, the country I live in faced record fires in 2019/20 and now floods, though this is happening all over the globe. The evidence is in our face, but people are still in denial. It takes a lot of cognitive dissonance to admit when we’re wrong, and that’s part of the problem.

 

Philosophy teaches you that you can have a range of views on a specific topic, and as I keep saying: only future generations know how ignorant the current generation is. That includes me, of course. I write a blog, which hopefully outlives me and one day people should be able to tell where I was wrong. I’m quite happy for that, because that’s how knowledge grows and progresses.

Beliefs, prejudices and theories; where is truth?

 During COVID, New Scientist started doing a lot of online events, including courses and ‘talks’ by experts in various fields. I watched one of these talks last week by Claudia De Rahm titled What We Don’t Know About Gravity, which I thought was very good. It was informative and thought-provoking and therefore deserves special mention. Claudia is a young woman, Professor of Physics at Imperial College London, with a distinct Italian accent. She gestures a lot while she’s talking and exudes passion. Sometimes her face appeared childlike, especially at the end when she conveyed her appreciation to the presenter, Martin Davies. She’s won a number of awards and she’s done research in particle physics, gravity and cosmology.

 

One of the first things she told us is that Einstein’s GR (general theory of relativity) comes with its own ‘proof’ of its limitations. She didn’t use the word proof, but she demonstrated what she meant. If one tries to apply QM to Einstein’s mathematical theory you get probabilities of over 100%. I never knew this, but I found it a remarkable revelation. From what I could gather, it happens near the Planck scale where the curvature of spacetime becomes so large the physics breaks down. She pointed out that this doesn’t occur near the event horizon of a black hole, so for everything we can observe, GR is perfectly valid. But I was astounded to learn that GR predicts its own failure at certain scales of the Universe.

 

She also questioned whether GR breaks down at the other extreme of scale, given that there is disagreement on how fast the Universe is expanding to a significant degree (in her own words, ‘the chance of it being a fluke is 1 part in 14,000’). Of course, she also explained how 95% of the Universe is ‘missing’, meaning it can’t be accounted for. Personally, I think we’re ripe for another scientific revolution comparable to the one that occurred 100 years ago, which in turn was comparable to the one created by Copernicus, Galileo, Kepler and Newton.

 

This highlights a point I’ve raised before: the significance of scale in determining which ‘natural laws’ dominate, though they all seem to obey a Lagrangian (based on my limited knowledge of physics). Roger Penrose argues that scale is dependent on mass. If the Universe was all radiation then scale becomes irrelevant. This is essential for his CCC (Conformal Cyclic Cosmology) model of the Universe to work. Penrose also argues that there is no time without mass, because time is always zero for a photon. This creates a paradox, because the photon has an energy dependent on its frequency, which has no meaning without time. I’ve no doubt Penrose can resolve that, but I don’t know how. Perhaps gravity resolves that conundrum. But, as the Universe exists in its current epoch within our range of observations, scale plays a significant, even critical role in determining which mathematical formulations we use to model it.

 

Claudia tip-toed around the argument about whether gravity is a force or not, but gave me the impression she believes it isn’t. She did point out that a gravitational wave effectively creates a force and there are tidal forces, but this is not what people mean when they argue that there is a ‘force of gravity’ in the Newtonian sense. In answer to a question at the end, she pointed out that “gravity is related to the very structure of spacetime; you can never switch it off”.

 

On the subject of GR’s inherent limitations around a singularity inside a black hole, she seemed optimistic that new physics would overcome this eventually. Along with the questions around dark energy and dark matter, that comprise 70% and 25% of the Universe respectively, I think that only a revolution in physics and cosmology will rescue it. Towards the end of the talk, she put up a slide showing all the current theories in the running, without discussing any of them or mentioning any personal favourites she might have. She literally covered the screen with balloons of speculative ideas, demonstrating the burgeoning interest in this field.

 

And this segues into something else she said in answer to a question, where someone asked if all the current theories should be ‘thrown in the rubbish bin’ and replaced with something completely different. She pointed out that the current theories work extremely well, and whatever you replace them with has to, at the very least, account for what we already ‘know’, and you can’t just ‘throw them in the rubbish bin’. This touches on the subject of my last post where people sometimes argue that we really don’t ‘know’ anything and we only have 'beliefs'. In science, all theories have limitations. Truth is cumulative in science; just because we don’t know everything, it doesn’t mean that what we do know is wrong and should be thrown out. Personally, I don’t think there will ever be a TOE (theory of everything) simply because there’s never been one in the past, and people have always ‘believed’ that we know almost everything, which history has proved, repeatedly, is untrue.

 

And this brings me to the subject of pet theories or pet prejudices. If Claudia has her own pet theories she didn’t elaborate, yet I’m sure she has. People much smarter than me have their pet prejudices, some of which differ dramatically, so they can’t all be right, and that also applies to me. But, having said that, I like to think my prejudices are well informed and I acknowledge those who share them, and sometimes those who don’t.

 

I will quickly talk about one that is relevant and that is time. I contend that consciousness exists in a constant present, while everything we observe has already happened, which is why we ‘feel’ like we’re travelling through time. According to relativity theory, we are travelling through time just by standing still. But when we move, we start travelling through space and, as a consequence, we travel through time more slowly – that is, time slows down. In fact, if we could travel through space at the speed of light, we would stop travelling through time altogether. But here’s the thing: that’s only true in our specific frame of reference. There could be another frame of reference, like the horizon of the observable universe where space itself travels at the speed of light. I discussed this in another post.

 

This infers that everything travels through time and not just consciousness. However, while our consciousness remains in a constant present, our thoughts don’t. Our thoughts become memories as soon as we think them, otherwise we wouldn’t even know we think. Consciousness exists on the edge of time and so does the universe itself. I’ve no reason to believe that the edge of time we all experience isn’t concordant with the edge of time for the whole Cosmos. This is considered naive thinking, but it’s one of my pet prejudices.