Symptoms of living in a post-truth world

 I recently had 2 arguments with different people, who took extreme positions on what we mean by truth. One argued that there is no difference between mathematical truths and truths in fiction – in fact, he described mathematics, that is not being ‘applied’, as ‘mathematical fiction’. The other argued that there is no objective truth and everything we claim to know are only ‘beliefs’, including mathematics. When I told her that I know there will always be mathematics that remain unknown, she responded that I ‘believe I know’. I thought that was an oxymoron, but I let it go. The trivial example, that there are an infinite number of primes or an infinite number of digits in pi, should put that to rest, or so one would think. 

Norman Wildberger, whom I’ve cited before, says that he doesn’t ‘believe’ in Real numbers, and neither does he believe in infinity, and he provides compelling arguments. But I feel that he’s redefining what we mean by mathematics, because his sole criterion is that it can be computed. Meanwhile, we have a theorem by Gregory Chaitin who contends that there are infinitely more incomputable Real numbers than computable Real numbers. People will say that mathematics is an abstract product of the mind, so who cares. But, as Paul Davies says, ‘mathematics works’, and it works so well that we can comprehend the Universe from the cosmic scale to the infinitesimal. 

 

Both of my interlocutors, I should point out, were highly intelligent, well-educated and very articulate, and I believe that they really believed in what they were saying. But, if there is no objective truth, then there are no 'true or false' questions that can be answered. To take the example I’ve already mentioned, it’s either true or false that we can’t know everything in mathematics. And if it’s false, then we must know everything. But my interlocutor would say that I claimed we’d never know and I can’t say I know that for sure. 

 

Well, putting aside the trivial example of infinity, there are proofs based on logic that says it’s true and that’s good enough for me. She claimed that logic can be wrong if the inputs are wrong, which is correct. In mathematics, this is dependent on axioms, and mathematics like all other sciences never stands still, so we keep getting new axioms. But it’s the nature of science that it builds on what went before, and, if it’s all ‘belief’, then it’s a house built on sand. And if it's a house built on sand, then all the electronic gadgets we use and the satellite systems we depend on could all crash without warning, but no one really believes that.

 

So that’s one side of the debate and the other side is that truths in art have the same status as truths in science. There are a couple of arguments one can use to counter this, the most obvious being that a work of art, like Beethoven’s 5^th, is unique – no one else created that. But Pythagoras’s theorem could have been discovered by anyone, and in fact, it was discovered by the Chinese some 500 years before Pythagoras. I write fiction, and while I borrow tropes and themes and even plot devices from others, I contend that my stories are unique and so are the characters I create. In fact, my stories are so unique, that they don’t even resemble each other, as at least one reader has told me.

 

But there is another argument and that involves memes, which are cultural ideas, for want of a better definition, that persist and propagate. Now, some argue that virtually everything is a meme, including scientific theories and mathematical theorems. But there is a difference. Cultural memes are successful because they outlive their counterparts, but scientific theories and mathematical theorems outlive their counterparts because they are successful. And that’s a fundamental distinction between truth in mathematics and science, and truth in art.

Addendum: I just came across this video (only posted yesterday) and it’s very apposite to this post. It’s about something called Zero Knowledge Proof, and it effectively proves if someone is lying or not. It’s relevance to my essay is that it applies to true or false questions. You can tell if someone is telling the truth without actually knowing what that truth is. Apparently, it’s used algorithmically as part of blockchain for bitcoin transactions.

 

To give the example that Jade provides in her video, if someone claims that they have a proof of Riemann’s hypothesis, you can tell if they’re lying or not without them having to reveal the actual proof. That’s a very powerful tool, and, as a consequence, it virtually guarantees that a mathematical truth exists for a true or false proposition; in this hypothetical case, Riemann’s hypothesis, because it’s either true or false by definition.

We are not just numbers, but neither is the Universe

 A few years back I caught up with someone I went to school with, whom I hadn’t seen in decades, and, as it happened, had studied civil engineering like me. I told him I had a philosophy blog where I wrote about science and mathematics, among other things. He made the observation that mathematics and philosophy surely couldn’t be further apart. I pointed out that in Western culture they had a common origin, despite a detour into Islam, where mathematics gained a healthy and pivotal influence from India. 

I was reminded of this brief exchange when I watched this Numberphile video on the subject of numbers, where Prof Edward Frankel (UC Berkeley) briefly mentions the role of free will in our interaction with mathematics.

 

But the main contention of the video is that numbers do not necessarily have the status that we give them in considering reality. In fact, this is probably the most philosophical video I’ve seen on mathematics, even though Frankel is not specifically discussing the philosophy of mathematics.

 

He starts off by addressing the question whether our brain processes are all zeros and ones like a computer, and obviously thinks not. He continues that in another video, which I might return to later. The crux of this video is an in-depth demonstration of how a vector can be represented by a pair of numbers. He points out that the numbers are dependent on the co-ordinate system one uses, which is where ‘free will’ enters the discussion, because someone ‘chooses’ the co-ordinate system. He treats the vector as if it’s an entity unto itself, which he says ‘doesn’t care what co-ordinates you choose’. Brady, who is recording the video, takes him up on this point: that he’s effectively personifying the vector. Frankel acknowledges this, saying that it’s an ‘abstraction within an abstraction.’

 

Now, Einstein used vectors in his general theory of relativity, and one of the most important points was that the vectors are independent of the co-ordinate system. So we have this relationship between a mathematical abstraction and physical reality. People often talk about mistaking the ‘map for the terrain’ and Frankel uses a different metaphor where he says, ‘don’t confuse the menu for the meal’. I agree with all this to a point.

 

My own view is that there are 2 aspects of mathematics that are conflated. There is the language of mathematics, which includes the numbers and the operations we use, and which are ‘invented’ by humans. Then there are the relationships, which this language describes, but which are not prescribed by us. There is a sense that mathematics takes on a life of its own, which is why Frankel can talk about a vector as if it has an independent existence to him. Then there is Einstein who incorporated vectors into his mathematical formulation to describe how gravity is related to spacetime. 

 

Now here’s the thing: the relationship between gravity and spacetime still exists without humans to discover it or describe it. Spacetime is the 3 dimensions of space and 1 of time that, along with gravity, allows planets to maintain orbits over millions of years. But here’s the other thing: without mathematics, we would never know that or be able to describe it. It’s why some claim that mathematics is the language of nature. Whether Frankel agrees or not, I don’t know.

 

In the second video, Brady asks Frankel if he thinks he’s above mathematics, which makes him laugh. What Frankel argues is that there are inner emotional states, like ‘falling in love’, which can’t be described by mathematics. I know that some people would argue that falling in love is a result of biochemical algorithms, nevertheless I agree with Frankel. You can construct a computer model of a hurricane but it doesn’t mean that it becomes one. And it’s the same with the brain. You might, as someone aspired to do, create a computer model of a human brain, but that doesn’t mean it would think like one.

 

This all brings me back to Penrose’s 3 worlds philosophy of the mathematical, the mental and the physical and their intrinsic relationships. In a very real way, numbers allow us to comprehend the physical world, but it is not made of numbers as such. Numbers are the basis of the language we use to access mathematics, because I believe that’s what we do. I’ve pointed out before, that equations that describe the physical world (like Einstein’s) have no meaning outside the Universe, because they talk about physical entities like space and time and energy – things we can measure, in effect.

 

On the other hand, there are mathematical relationships, like Riemann’s hypothesis, for example, that deals with an infinity of primes, which literally can’t be contained by the Universe, by definition. At the end of the 2^ndvideo, Frankel quickly mentions Godel’s Incompleteness Theorem, which he describes in a nutshell by saying that there are truths in mathematics that can’t be formally proven. So there is a limit on what the human mind can know, given a finite universe, yet the human mind is 'not a mathematical machine’, as he so strongly argues.

 

He discusses more than I’ve covered, like his contention that our fixation with the rational is ‘irrational’, and there is no proof for the existence or non-existence of God. So, truly philosophical.

Natural laws; a misnomer?

I’ve referenced Raymond Tallis before, and I have to say up front that I have a lot of respect for his obvious erudition and the breadth of his intellectual discourse. He is an author and regular columnist in Philosophy Now, with a background in neuroscience. I always read his column, because he’s erudite and provocative. In Issue 144 (June/July 2021) he wrote an essay titled, The Laws of Nature. He didn’t use the term ‘misnomer’ anywhere, but that was the gist of his argument.

Tallis and I have a fundamental disagreement concerning the philosophy of science; and physics, in particular. This will become obvious as I expound on his article. He starts by pointing out how the word ‘law’ has theological connotations, as well as cultural ones. It’s a word normally associated with humanmade rules or edicts, which are necessary just so we can live together. An obvious one is what side of the road to drive on, otherwise we would have carnage and road-rage would be the least of our worries.

 

Science evolved out of a religious epistemology (I know that’s an oxymoron), but the pioneers of physics, like Galileo, Kepler and Newton, were all religious people and, from their perspective, they were uncovering ‘God’s laws’. This even extended to Einstein, who often referred to ‘God’ in a metaphorical sense, and saw himself and his contemporary physicists as uncovering the ‘Old One’s Secrets’. Even Stephen Hawking, a self-declared atheist, coined the phrase, ‘The Mind of God’.

 

So I agree with Tallis on this point that the use of the word, law, in this context, is misleading and carries the baggage of an earlier time, going back to the ancient Greeks (and other cultures) that human affairs were contingent on the whims of the Gods.

 

So Tallis searched around for an alternative term, and came up with ‘habits’, whilst admitting that it’s not ideal and that ‘it will have to punch above its usual weight’. But I think Tallis chose the word because, in human terms, ‘habit’ means something we acquire out of familiarity, and may or may not be the best method, or approach, to a specific situation. The idea that nature follows ‘habits’ implies there is no rhyme or reason behind their efficacy or apparent success. Even the word, success, is loaded, yet I think it subverts his point, because they are ‘successful’ in the sense that they ultimately produced a lifeform that can cognise them (more on that below).

 

Tallis makes the point that in nature ‘things just happen’, and the ‘laws’ are our attempt to ‘explain’ them. But, extending this line of thought, he suggests that actually we invent laws to ‘describe’ what nature does, which is why ‘habits’ is a better term.

 

The expectation of finding an explanation of nature’s regularity is the result of extrapolating to the whole of things the belief that every individual thing happens for a reason – that nothing ‘just happens’.

 

The word ‘regularity’ is apt and is one that physicists often use, because that is what we have learned about nature on all scales, and it is why it is predictable to the degree that it is. There is, of course, a missing element in all this, and that is the role of mathematics. I’m not surprised that Tallis doesn’t mention the word (even once as best I can tell), because he believes that physicists have a tendency to ‘mistake the map for the territory’ when they invoke mathematics as having a pivotal role in our epistemology. In another essay, he once argued that the only reason mathematics has a place in physics is because we need to measure things, or quantify them, in order to make predictions that can be verified. However, the very laws (or habits) that are the subject of his essay, are completely dependent on mathematics to be comprehensible at all.

 

In closing, Tallis makes a very good argument: there is a gap between the ‘habits’ that nature follows and the humanmade ‘laws’ in our science that we use to describe these habits. He makes the point that we are forever trying to close this gap as we discover more about nature’s habits. And he’s right, because it appears that, no matter how much we learn, there are always more of nature’s secrets to decipher. Every theory we’ve devised thus far has limits and we’ve even reached a point where our theory for the very large appears irreconcilable, mathematically, with our theory for the very small. But the point I’d make is that mathematics not only gives us our best description of reality, it also delineates the limitations of any particular theory. Consequently, I contend there will always be a gap.

 

Physicists say that the best we can do is provide a model and that model is always mathematical. Hawking made this point in his book, The Grand Design. So the model describes the laws, or habits, to the extent that we understand them at the time, and that it gets updated as we learn more.

 

Tallis mentions the well-known example of Newton’s ‘laws’ being surpassed by Einstein’s. But here’s the thing: the ‘inverse square law’ still applies and that’s not surprising, as it’s dependent on the Universe existing in 3 spatial dimensions. So we not only have a ‘law’ that carries over, but we have an explanation for it. But here’s another thing: the 3 spatial dimensions in combination with the single dimension of time is probably the only combination of dimensions that would allow for a universe to be habitable. Cosmologist and Fellow of the Royal Society, John D Barrow, expounds on this in some detail in his book, The Constants of Nature. (As a side note, planets can only remain in stable orbits over astronomical time periods in 3 dimensions of space.) So where I depart philosophically from Tallis, is that there are fundamental parameters in the Universe’s very structure that determine the consequences of something existing that can understand that structure. 

 

Nevertheless, I agree with Tallis to the extent that I think the term, law, is a misnomer, and I think a better word is ‘principle’. If one goes back to Einstein’s theory of gravity replacing Newton’s, it introduces a fundamental principle called the 'principle of least action', which I think was pointed out by Emmy Noether, not Einstein. As it turns out, the principle of least action also ‘explains’ or ‘describes’ optical refraction, as well as forming the basis of Richard Feynman’s path integral method for QED (quantum electrodynamics). The principle of least action, naturally, has a mathematical formulation called the Lagrangian.

 

Speaking of Emmy Noether, she derived a famous mathematical theorem (called Noether’s theorem) that is a fundamental ‘principle’ in physics, describing the intrinsic relationship between symmetries and conservation laws. It’s hard to avoid the term, law, in this context because it appears to be truly fundamental based on everything we know.

 

So, is this a case of confusing the map with the terrain? Maybe. The Universe doesn’t exist in numbers – it exists as a process constrained by critical parameters, all of which can only be deciphered by mathematics. To give just one example: Planck’s constant, h, determines the size of atoms which form the basis of everything you see and touch.

Other relevant posts: the-lagrangian-possibly-most.html

                                   the-universes-natural-units_9.html

Monty Hall Paradox explained

This is a well known problem based on a 1960s US television game show called Let’s Make a Deal. How closely it resembles that particular show, I don’t know, but it’s not relevant, because it’s easy to imagine. The show’s host’s stage-name was Monty Hall, hence the name of the puzzle.

 

In 1975, an American statistician and professor at the University of California, Berkeley, Steve Selvin, published a short article on the Monty Hall Paradox in The American Statistician, which he saw as a curiosity for a very select group who would appreciate its quirkiness and counter-intuitive answer. He received some criticism, which he easily countered.

 

Another totally unrelated (weekly) periodical, Parade magazine, with a circulation in the tens of millions, had a column called Ask Marilyn, who specialised in solving mathematical puzzles, brain teasers and logical conundrums sent to her by readers. She was Marilyn vos Savant, and entered the Guinness Book of Records in the 1980s as the woman with the highest recorded IQ (185). I obtained all this information from Jim Al-Khalili’s book, Paradox; The Nine Greatest Enigmas in Physics.

 

Someone sent Marilyn the Monty Hall puzzle and she came up with the same counter-intuitive answer as Selvin, but she created an uproar and was ridiculed by mathematicians and academics across the country. Al-Khalili publishes a sample of the responses, at least one of which borders on misogynistic. Notwithstanding, she gave a more comprehensive exposition in a later issue of Parade, emphasising a couple of points I’ll come to later.

 

Now, when I first came across this puzzle, I, like many others, couldn’t understand how she could possibly be right. Let me explain.

 

Imagine a game show where there is just a contestant and the host, and there are 3 doors. Behind one of the doors is the key to a brand-new car, and behind the other 2 doors are goats (pictures of goats). The host asks the contestant to select a door. After they’ve made their selection, the host opens one of the other 2 doors revealing a goat. Then he makes an offer to the contestant, saying they can change their mind and choose the other door if they wish. In the original scenario, the host offers the contestant money to change their mind, upping the stakes.

 

Now, if you were a contestant, you might think the host is trying to trick you out of winning the car (assuming the host knows where the car is). But, since you don’t know where the car is, you now have a 1 in 2 chance of winning the car, whereas before you had a 1 in 3 chance. So changing doors won’t make any difference to your odds.

 

But both Selvin and vos Savant argued that if you change doors you double your chances. How can that be?

 

I found a solution on the internet by the Institute of Mathematics, giving a detail history and a solution using Bayes’ Theorem, which is difficult to follow if you’re not familiar with it. The post also provides an exposition listing 5 assumptions. In common with Al-Khalili, the author (Clive Rix from the University of Leicester), shows how the problem is similar to one posed by Martin Gardner, who had a regular column in Scientific American, involving 3 prisoners, one of whom would be pardoned. I won’t go into it, but you can look it up, if you’re interested, by following the link I provided.

 

What’s important is that there are 2 assumptions that change everything. And I didn’t appreciate this until I read Al-Khalili’s account. Nevertheless, I found it necessary to come up with my own solution.

 

The 2 key assumptions are that the host knows which door hides the car, and the host never picks the car.

 

So I will describe 3 scenarios:

 

1)    The assumptions don’t apply.

2)    We apply assumption No1.

3)    We apply assumptions 1 & 2.

 

In all 3 scenarios, the contestant chooses first.

 

In scenario 1: the contestant has a 1 in 3 chance of selecting the car. If the contest is run a number of times (say, 100 or so), the contestant will choose the car 1/3 of the times, and the host will choose the car 1/3 of the time, and 1/3 of the time it’s not chosen by either of them.

 

Scenario 2: the host knows where the car is, but he lets the contestant choose first. In 1/3 of cases the contestant chooses the car, but now in 2/3 of cases, the host can choose the car.

 

Scenario 3: the host knows where the car is and never chooses the car. Again, the contestant chooses first and has a 1 in 3 chance of winning. But the host knows where the car is, and in 1/3 of cases it's like scenario 1. However, in 2/3 of cases he chooses the door which doesn’t have the car, so the car must be behind the other door. Therefore, if the contestant changes doors they double their chances from 1 in 3 to 2 in 3.

The dialectic between science and philosophy

 This is a consequence of a question I answered on Quora. It was upvoted by the person who requested it, which is very rare for me (might have happened once before).

Naturally, I invoke one of my favourite metaphors. Contrary to what some scientists claim (Stephen Hawking comes to mind) philosophy is not dead, and in fact science and philosophy have a healthy relationship.

 

The original question was:

 

I see a distinction between "Philosophy" which attempts to describe and explain the world and "Science" in which theories can be proven or disproven. Under this paradigm, what areas claiming to be Science are actually Philosophy? (Requested by Michael Wayne Box.)

 

There are, in fact, 98 answers to this question, many by academics, which makes my answer seem pretty pedestrian. But that’s okay; from what I've read, I took a different approach. I’m interested in how science and philosophy interact (particularly epistemology) rather than how they are distinct, though I address that as well. My answer below:

 

 

Science as we know it today is effectively a product of Western philosophy going all the way back to Plato’s Academy and even earlier. In fact, science was better known as ‘natural philosophy’ for much of that time.

 

Science and epistemology have a close relationship, and I would argue that it is a dialectical relationship. John Wheeler provides a metaphor that I find particularly appealing:

 

We live on an island of knowledge in a sea of ignorance. As the island of knowledge expands, so does the shoreline of our ignorance.

 

I came up with this metaphor myself, independently of Wheeler, and I always saw the island of knowledge as science and the shoreline as philosophy. Seen in this light, I contend that science and philosophy have a dialectical relationship. But there is another way of looking at it, and I’d like to quote Russell:

 

Thus, to sum up our discussion of the value of philosophy; Philosophy is to be studied, not for the sake of any definite answers to its question, since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conception of what is possible, enrich our intellectual imagination and diminish the dogmatic assurance which closes the mind against speculation; but above all because, through the greatness of the universe, which philosophy contemplates, the mind is also rendered great, and becomes capable of that union with the universe which constitutes its highest good.’ (My emphasis)

 

And this, I believe, touches on the main distinction between science and philosophy: that philosophy asks questions for which we currently don’t have answers and science actually provides answers, even if they’re provisional. And that’s why I argue there is a dialectic, because, as soon as science gives us an answer, it also gives us new questions.

 

To provide examples: Do we live in a multiverse? Will AI become sentient? Was there something before the Big Bang? What is dark matter? How did DNA evolve? All these questions are on the shoreline of our island of knowledge. They are at the boundary between philosophy and science. But that boundary changes as per Wheeler’s metaphor.

 

 

 

That’s what I wrote on Quora, but I can’t leave this subject without talking about the role of mathematics. Curiously, mathematics is also linked to philosophy via Plato and his predecessors. So I will add a comment I wrote on someone else’s post. It might be hubris on my part, but 20^th Century’s preoccupation with language defining our epistemology seems to miss the point, because the Universe speaks to us in its own language, which is mathematical. 

 

With that in mind, this is what I wrote in response to a post that concluded, Philosophy will continue as it has for the past two and a half thousand years, but we presently stand over the wreckage of a philosophical tradition four centuries in the making.

 

 

 

I believe philosophy is alive and well; it’s just had a change of clothes. You can’t divorce philosophy from science, with which it has a dialectical relationship. And that’s just concerning epistemology and ontology.

 

The mistake made in the early 20th Century was to believe mathematics is an artefact created by logic, when in fact, logic is what we use to access mathematics. Godel’s theorem and Turing’s resolution to the halting problem, demonstrate that there will always exist mathematical ‘truths’ that we can’t prove (even if we prove them there will be others). In effect, Godel ‘proved’ (ironically) that there is a difference between ‘truth’ and ‘proof’ in mathematics.

 

But there’s more: Einstein and his cohorts involved in the scientific revolution that took up the whole 20th Century, showed that mathematics lies at the heart of nature. To quote Richard Feynman:

 

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.

 

So we have an epistemological link between the natural world and mathematics. It’s not human language that defines what we call reality, but mathematics. From what I read on this subject, it seems philosophers in general still haven’t caught up with this truth.

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.

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.

Is the Universe deterministic?

 I’ve argued previously, and consistently, that the Universe is not deterministic; however, many if not most physicists believe it is. I’ve even been critical of Einstein for arguing that the Universe is deterministic (as per his famous dice-playing-God statement). 

Recently I’ve been watching YouTube videos by theoretical physicist, Sabine Hossenfelder, and I think she’s very good and I highly recommend her. Hossenfelder is quite adamant that the Universe is deterministic, and her video arguing against free will is very compelling and thought-provoking. I say this, because she addresses all the arguments I’ve raised in favour of free will, plus she has supplementary videos to support her arguments.

 

In fact, Hossenfelder states quite unequivocally towards the end of the video that ‘free will is an illusion’ and, in her own words, ‘needs to go into the rubbish bin’. Her principal argument, which she states right at the start, is that it’s ‘incompatible with the laws of nature’. She contends that the Universe is completely deterministic right from the Big Bang. She argues that everything can be described by differential equations, including gravity and quantum mechanics (QM), which she expounds upon in some detail in another video. 

 

My immediate reaction to this: is what about Poincare and chaos theory? Don’t worry, she addresses that as well. In fact, she has a couple of videos on chaos theory (though one is really about weather and climate change), which I’d recommend.

 

The standard definition of chaos is that it’s deterministic but unpredictable, which seems to be an oxymoron. As she points out, chaotic phenomena (which includes the weather and the orbits of the planet, among many other things, like evolution) are dependent on the ‘initial conditions’. An infinitesimal change in the initial conditions will result in a different outcome. The word ‘infinitesimal’ is the key here, because you need to work out the initial conditions to an infinite decimal place to get the answer. That’s why it’s not predictable. As to whether it’s deterministic, I think that’s another matter.

 

To overcome this apparent paradox, I prefer to say it’s indeterminable, which is not contentious. Hossenfelder explains, using a subtly different method, that you can mathematically prove, for any chaotic system, that you can only forecast to a finite time in the future, no matter how detailed your calculation (it’s worth watching her video, just to see this).

 

Because the above definition for chaos seems to lead to a contradiction or, at best, an oxymoron, I prefer another definition that is more pragmatic and is mostly testable (though not always). Basically, if you rerun a chaotic phenomenon, you’ll get a different outcome. The best known example is tossing a coin. It’s well known in probability theory (in fact it’s an axiom) that the result of the next coin toss is independent of all coin tosses that may have gone before. The reason for this is that coin tosses are chaotic. The same principle applies to throwing dice, and Marcus du Sautoy expounds on the chaos of throwing dice in this video. So, tossing coins and throwing dice are considered ‘random’ events in probability theory, but Hossenfelder contends they are totally deterministic; just unpredictable.

 

Basically, she’s arguing that just because we can’t calculate the initial conditions, they still happened and therefore everything that arises from them is deterministic. Du Sautoy (whom I referenced above) in the same video and in his book, What We Cannot Know, cites physicist turned theologian, John Polkinghorne, that chaos provides the perfect opportunity for an interventionist God – a point I’ve made myself (though I’m not arguing for an interventionist God). I’m currently reading Troy by Stephen Fry, an erudite rendition based on Homer’s tale, and it revolves around the premise that one’s destiny is largely predetermined by the Gods. The Hindu epic, Mahabharata, also portrays the notion of destiny that can’t be avoided. Leonard Cohen once remarked upon this in an interview, concerning his song, If It Be Your Will. In fact, I contend that you can’t believe in religious prophecy if you don’t believe in a deterministic universe. My non-belief in a deterministic universe is the basis of my argument against prophecy. And my argument against determinism is based on chaos and QM (which I’ll come to shortly).

 

Of course, one can’t turn back the clock and rerun the Universe, and, as best I can tell, that’s Hossenfelder’s sole argument for a deterministic universe – it can’t be changed and it can’t be predicted. She mentions Laplace’s Demon, who could hypothetically calculate the future of every particle in the Universe. But Laplace’s Demon is no different to the Gods of prophecy – it can do the infinite calculation that we mortals can’t do.

 

I have to concede that Hossenfelder could be right, based on the idea that the initial conditions obviously exist and we can’t rewind the clock to rerun the Universe. However, tossing coins and throwing dice demonstrate unequivocally that chaotic phenomena only become ‘known’ after the event and give different outcomes when rerun. 

 

So, on that basis, I contend that the future is open and unknowable and indeterminable, which leads me to say, it’s also non-deterministic. It’s a philosophical position based on what I know, but so is Hossenfelder’s, even though she claims otherwise: that her position is not philosophical but scientific.

 

Of course, Hossenfelder also brings up QM, and explains it is truly random but it’s also time reversible, which can be demonstrated with Schrodinger’s equation. She makes the valid point that the inherent randomness in QM doesn’t save free will. In fact, she says, ‘everything is either determined or random, neither of which are affected by free will’. However, she makes the claim that all the particles in our brain are quantum mechanically time reversible and therefore deterministic. However, I contend that the wave function that allows this time reversibility only exists in the future, which is why it’s never observed (I acknowledge that’s a personal prejudice). On the other hand, many physicists contend that the wave function is a purely mathematical construct that has no basis in reality.

 

My argument is that it’s only when the wave function ‘collapses’ or ‘decoheres’ that a ‘real’ physical event is observed, which becomes classical physics. Freeman Dyson argued something similar. Like chaotic events, if you were to rerun a quantum phenomenon you’d get a different outcome, which is why one can only deal in probabilities until an ‘observation’ is made. Erwin Schrodinger coined the term ‘statistico-deterministic’ to describe QM, because at a statistical level, quantum phenomena are predictable. He gives the example of radioactive decay, which we can predict holistically very accurately with ‘half-lives’, but you can’t predict the decay of an individual isotope at all. I argue that, both in the case of QM and chaos, you have time asymmetry, which means that if you could hypothetically rewind the clock before the wave function collapse or some initial conditions (whichever the case), you would witness a different outcome.

 

Hossenfelder sums up her entire thesis with the following statement:

 

...how ever you want to define the word [free will], we still cannot select among several possible different futures. This idea makes absolutely no sense if you know anything about physics.

 

Well, I know enough about physics to challenge her inference that there are no ‘possible different futures’. Hossenfelder, herself, knows that alternative futures are built-into QM, which is why the multiple worlds interpretation is so popular. And some adherents of the Copenhagen interpretation claim that you do get to ‘choose’ (though I don’t). If the wave function describes the future, it can have a multitude of future paths, only one of which becomes reality in the past. This derives logically from Dyson’s interpretation of QED.

 

Of course, none of this provides an argument for free will, even if the Universe is not deterministic.

 

Hossenfelder argues that the brain’s software (her term) runs calculations that determine our decisions, while giving the delusion of free will. I thought this was her best argument:

 

Your brain is running a calculation, and while it is going on you do not know the outcome of that calculation. So the impression of free will comes from our ‘awareness’ that we think about what we do, along with our inability to predict the result of what we are thinking.

 

You cannot separate the idea of free will from the experience of consciousness. In another video, Hossenfelder expresses scepticism at all the mathematical attempts to describe or explain consciousness. I’ve argued previously that if we didn’t all experience consciousness, science would tell us that it is an illusion just like free will is. That’s because science can’t explain the experience of consciousness any better than it can explain the intuitive sense of free will that most of us take for granted.

 

Leaving aside the use of the words, ‘calculation’ and ‘software’, which allude to the human brain being a computer, she’s right that much of our thinking occurs subconsciously. All artists are aware of this. As a storyteller, I know that the characters and their interactions I render on the page (or on a computer screen) largely come from my subconscious. But everyone experiences this in dreams. Do you think you have free will in a dream? In a so-called ‘lucid dream’, I’d say, yes.

 

I would like to drop the term, free will, along with all its pseudo-ontological baggage, and adopt another term, ‘agency’. Because it’s agency that we all believe we have, wherever it springs from. We all like to believe we can change our situation or exert some control over it, and I’d call that agency. And it requires a conscious effort – an ability to turn a thought into an action. In fact, I’d say it’s a psychological necessity: without a sense of agency, we might as well be automatons.

 

I will finish with an account of free will in extremis, as told by London bomber survivor, Gill Hicks. Gill Hicks was only one person removed from the bomber in one of the buses, and she lost both her legs. As she tells it, she heard a voice, like we do in a dream, and it was a female voice and it was ‘Death’ and it beckoned to her and it was very inviting; it was not tinged with fear at all. And then she heard another voice, which was male and it was ‘Life’, and it told her that if she chose to live she had a destiny to fulfil. So she had a choice, which is exactly how we define free will and she consciously chose Life. As it turned out, she lost 70% of her blood and she had a hole in the back of her head from a set of keys. In the ambulance, she later learned that she was showing no signs of life – no pulse and she had flatlined – yet she was talking. The ambo told the driver, ‘Dead but talking.’ It was only because she was talking that he continued to attempt to save her life.

 

Now, I’m often sceptical about accounts of ‘near-death experiences’, because they often come across as contrived and preachy. But Gill Hicks comes across as very authentic; down-to-Earth, as we say in Oz. So I believe that what she recalled is what she experienced. I tell her story, because it represents exactly what Hossenfelder claims about free will: it defies a scientific explanation.