The short answer is no; there is no ‘fire in the equations’. One needs to be careful not to conflate epistemology with ontology. Let’s look at the wave function (ψ) which is a fundamental entity in quantum mechanics (QM). It’s a mathematical formula that gives probabilities of finding a particle existing before the particle is actually ‘observed’. However, there is also some debate about whether the wave function exists in reality.
Mathematics, from a human perspective, is a set of symbols that can be arranged in formulae that can describe and predict physical phenomena. The symbols are human-made, but the relationships, that are entailed in the formulae, are not. In other words, mathematical relationships appear to have a life of their own independent of human minds.
So there is a relationship between mathematics, the physical world and the human mind, (probably best explored, if not explained, by Roger Penrose’s 3 worlds philosophy). The relationship between the human mind and the physical world is epistemological - epitomised by the discipline called physics. And mathematics is the medium we use in pursuing that epistemology.
Eugene Wigner famously wrote an essay called The unreasonable effectiveness of mathematics in the natural sciences, and it still causes debate half a century after it was written. Wigner refers to the 2 miracles inherent in the Universe’s capacity to be self-comprehending:
It is difficult to avoid the impression that a miracle confronts us here… or the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them.
Or to quote Einstein: The most incomprehensible thing about the Universe is that it’s comprehensible.
The point is that Wigner’s ‘miracles’ or Einstein’s ‘incomprehensible thing’ are completely dependent on mathematics. But Wigner, in particular, brings together epistemology and ontology under one rubric. Ontology is ‘the nature of being’ (dictionary definition). At its deepest level, the ‘nature of being’ appears to be mathematical.
None other than Richard Feynman weighed into the discussion in his book, The Character of Physical Law, specifically in a chapter titled The Relation of Mathematics to Physics, where he expounds:
...what turns out to be true is that the more we investigate, the more laws we find, and the deeper we penetrate nature, the more this disease persists. Every one of our laws is a purely mathematical statement in rather complex and abstruse mathematics... Why? I have not the slightest idea. It is only my purpose to tell you about this fact.
The ’disease’ he’s referring to and the ‘fact’ he can’t explain is best expressed in his own words:
The strange thing about physics is that for the fundamental laws we still need mathematics.
In conclusion, he says the following:
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.
Many scientists and philosophers argue that we create mathematical models that give very reliable and accurate descriptions of reality. All these ‘models’ have epistemological limits, which means we use different mathematics for different scenarios. Nevertheless, there are natural constants and mathematical ‘laws’ that are requisite for complex life to exist. Terry Bollinger (in a Quora post) explained the significance of Planck’s constant in determining the size and stability of atoms, from which everything we can see and touch is made, including ourselves. The fine structure constant is another fundamental dimensionless number that determines the ‘nature of being’ upon which the reality we all know depends.
So mathematics didn’t create the Universe, but, at a fundamental level, it determines the Universe we inhabit.
Footnotes:
1) This was in answer to a question posted on Quora. I did receive an 'upvote' from Masroor Bukhari, who is a former Research Fellow and PhD in Particle Physics at Houston University.
2) Will Singourd, who asked the question, wrote the following:
Thank you for that outstanding answer. This is the most thorough & best answer I've seen on Quora. I've printed it out for reference.
I appreciate all the thought you put in it, plus your elucidating writing skills.
A recurring theme on my blog has been the limits of what we can know. So Marcus du Sautoy’s book, What We Cannot Know, fits the bill. I acquired it after I saw him give a talk at the Royal Institute on the subject, promoting the book, which is entertaining and enlightening in and of itself. I’ve previously read his The Music of the Primes and Finding Moonshine, both of which are very erudite and stimulating. He’s made a few TV programmes as well.
Previously, I’ve written blog posts based on books by Bryan Magee (Ultimate Questions) and Noson S. Yanofsky (The Outer Limits of Reason; What Science, Mathematics, and Logic CANNOT Tell Us). Yanofsky is a Professor in computer science, while Magee was a Professor of Philosophy (later a broadcaster and Member of British Parliament). I have to admit that Yanofsky’s book appealed to me more, because it’s more science based. Magee’s book was very erudite and provocative; my one criticism being that he seemed almost dismissive of the role that mathematics plays in the limits of what we can know. He specifically states that “...rationality requires us to renounce the pursuit of proof in favour of the pursuit of progress.” (My emphasis). However, pursuit of proof is exactly what mathematicians do, and, what’s more, they do it consistently and successfully, even though there is a famous proof that says there are limits to what we can prove (Godel’s Incompleteness Theorem).
Marcus du Sautoy is a mathematician, and a very good communicator as well, as can be evidenced on some of his YouTube videos, including some with Numberphile. But his book is not limited to mathematics. In fact, he discusses pretty much all the fields of our knowledge which appear to incorporate limits, which he metaphorically calls ‘Edges’. These include, chaos theory, quantum mechanics, consciousness, the Universe, and of course, mathematics itself. One is tempted to compare his book with Yanofsky’s, as they are both very erudite and educational, whilst taking different approaches. But I won’t, except to say they are both worth reading.
One aspect of du Sautoy’s book, which is unusual, yet instructive, is that he consulted other experts in their respective fields, including John Polkinghorne, John Barrow, Kristof Koch and Robert May. May, in particular, did pioneering work in chaos theory on animal populations in the 1970s. An ex-pat Australian, he’s now a member of the House of Lords, which is where du Sautoy had lunch with him. All these interlocutors were very stimulating and worthy additional contributors to their respective topics.
Very early on (p.10, in fact) du Sautoy mentions a famous misprediction by French philosopher, Auguste Compte, in 1835, about the stars: “We shall never be able to study, by any method, their chemical composition or their mineralogical structure.” Yet, less than a century later, it was being done by spectroscopy as a virtually standard practice, which in turn led to the knowledge that the Universe was expanding consistently in all directions. Throughout the book, du Sautoy reminds us of Compte’s prediction, when it appears that there are some things we will never know. He also quotes Donald Rumsfeld on the very next page:
There are known knowns; these are things that we know that we know. We also know there are known unknowns, that is to say, we know there are some things we do not know. But there are also unknown unknowns, the ones we don’t know we don’t know.
At the time, people tended to treat Rumsfeld’s statement as a bit of a joke and a piece of political legerdemain, given its context: weapons of mass destruction. However, in the field of science, it’s perfectly correct: there are hierarchies of knowledge, and when one looks back, historically, there have always been unknown unknowns, and, therefore, it’s a safe bet they will exist in the future as well. In other words, our future discoveries are dependent on secrets the Universe has yet to reveal to us mere mortals.
Towards the end of his book, du Sautoy gets more philosophical, which is not surprising, and he makes a point that I’ve not seen or heard before. He argues that some things about the Universe, like time, and the possibility of a multiverse, might remain unknown without physically getting outside the Universe, which is impossible. This, of course, raises the issue of God. Augustine, among others, has argued that God exists outside the Universe, and therefore, outside time. Paul Davies made the same point in his book, The Mind of God, with specific reference to Augustine.
Du Sautoy, who is a self-declared atheist, contends that God represents what we cannot know, which is consistent with the idea that some things we cannot know, can only be known from outside the Universe. But du Sautoy makes the point that there is something that exists outside the Universe that we know and that is mathematics. He, therefore, makes the tongue-in-cheek suggestion that maybe we can replace God with mathematics. Curiously, John Barrow made the same mischievous suggestion in one of his books – probably, Pi in the Sky. According to du Sautoy, Barrow is a Christian, which surprised me as much as du Sautoy, given that you would never know it from his writings. While on the subject of God, John Polkinghorne is a well known theologian as well as a physicist. Again, according to du Sautoy, Polkinghorne contends that God could intervene in the Universe via chaos theory. I once made the same point, although I also said I didn’t believe in an interventionist God, as that leads to people claiming they know God’s will, and that leads to all sorts of acts done in God’s name, and we all know how that usually ends. The problem with believing in an interventionist God is that it axiomatically leads to people believing they can influence said God.
Getting back to the subject at hand, du Sautoy says:
If there was no universe, no matter, no space, nothing. I think there would still be mathematics. Mathematics does not require the physical world to exist.
Following on from du Sautoy’s book, I started re-reading Eli Maor’s book, e: the story of a number, which incidentally covers the history of calculus going back to the ancient Greeks and Archimedes, in particular. The Greeks had a problem in that they couldn’t acknowledge infinity – it was taboo. Maor believes that Archimedes must have known the concept of infinity because he appreciated how an iterative process could converge to a value, but he wasn’t allowed to say so. Even in the modern day, there are mathematicians who wish to be rid of the concept of infinity, yet it’s intrinsic to mathematics everywhere you look.
This is relevant because the very nature of infinity tells us that there will always be truths beyond our kin. You can use a Turing machine (a computer) to calculate all the zeros in Riemann’s hypothesis and, if it’s true, it will never stop. Now, du Sautoy makes an interesting observation about this (which he expounds upon in this video, if you want it firsthand) that it’s possible that Riemann’s hypothesis is unknowable. In fact, there’s a small collection of conjectures associated with prime numbers that fall into this category (the Goldbach conjecture and the twin-prime conjecture being another 2). But here’s the thing: if one can prove that the Riemann hypothesis is unknowable, then it must be true. This is because, if it was untrue, there would have to be at least one result that didn’t fit the hypothesis, which would make it ‘knowable’.
The unknowable possibility is a direct consequence of Godel’s Incompleteness Theorem. To quote du Sautoy:
Godel proved mathematically that within any axiomatic system framework for number theory that was free of contradictions there were true statements about numbers that could not be proved within that framework – a mathematical proof that mathematics has its limitations. (My empasis).
I highlighted that passage because I left it out when proposing a definition to someone on Quora, and as a consequence, my interlocutor tried to argue that my definition was incorrect. Basically, I was saying that within any axiomatic system of mathematics there are ‘truths’ that can’t be proven. That’s Godel’s famous theorem in essence and in practice. However, one can find proofs, in principle, by using new axioms outside that particular system. And we see this in practice. The axiom that geometry can be non-Euclidean created new proofs, and the introduction of √-1 created new mathematics, called complex algebra, that gave solutions to previously unsolvable problems.
Towards the end of his book, du Sautoy references a little known and obscure point made by the renowned logician Alonso Church, called the ‘paradox of unknowability’, which proves that unless you know it all, there will always be truths that are by their very nature unknowable.
In effect, Church has extended Godel’s theorem to the physical world. Du Sautoy gives the example of all the dice that are lost in his house. There is either an even number of them or an odd number. One of these is true, but it is unknowable unless he can find them all. A more universal example is whether the Universe is infinite or finite. One of these is true but it’s currently unknowable and may be for all time. Du Sautoy makes the point that if we learn it’s finite then it becomes knowable, but if it’s infinite it may remain forever unknowable. This is similar to the Riemann hypothesis being knowable or unknowable. If it’s false then the Turing machine stops, which makes it finite, but, if it’s true, it is both infinite and unknowable, based on that thought experiment. It was only at this point in my essay that I came up with its title. I’ve expressed it as a question, but it’s really a conclusion.
If we go back to Archimedes and his struggle with the infinite, we can see that probably for most of humankind’s history, the infinite was considered outside the mortal realm. In other words, it was the realm of God. In fact, du Sautoy quotes Descartes: God is the only thing I positively conceive as infinite.
I’ve long contended that mathematics is the only ‘realm’ (for want of a better word) where infinity is completely at home. In Maor’s book, at one point, he discusses the difference between applied mathematics and pure mathematics, and it occurred to me that this distinction could explain the perennial argument about whether mathematics is invented or discovered. But the plethora of infinities, which is also intrinsic to unknowable ‘truths’, as outlined above, infers that there will always be mathematical ‘things’ waiting to be discovered. What’s more, the ‘marriage’ between theoretical physics and pure mathematics has never been more productive.
Addendum 1: After writing this, I re-watched an interview with Norman Wildberger on the subject of infinity and Real numbers. Wildberger is an Australian mathematician with ‘unorthodox’ views on the foundations of mathematics, as he explains in the video.
Wildberger is not a crank: he’s an academic mathematician, who has unusual philosophical ideas on mathematics. He makes the valid point that computers can only work with finite numbers (meaning numbers with a finite decimal extension), and that is the criterion he uses to determine whether something mathematical is ‘real’. He says he doesn’t believe in Real numbers, as they are defined, because they are infinitely uncomputable.
In effect, he argues they have no place in the physical world, but I disagree. In chaos theory, the reason chaotic phenomena are unpredictable is because you have to calculate the initial conditions to infinite decimal places, which is impossible. This is both mathematical and physical evidence that some things are ‘unknowable’.
Addendum 2: Sabine Hossenfelder argues that infinity is only 'real' in the mathematical world. She contends that in physics, it's not 'real', because it's not 'measurable'. She gives a good exposition in this YouTube video.
I recently readThe Grand Designby Stephen Hawking (2010), co-authored by Leonard Mlodinow, who gets ‘second billing’ (with much smaller font) on the cover, so one is unsure what his contribution was. Having said that, other titles listed by Mlodinow (Euclid’s WindowandFeynman’s Rainbow) make me want to search him out. But the prose style does appear to be quintessential Hawking, with liberal lashings of one-liners that we’ve come to know him for. Also, I think one can confidently assume that everything in the book has Hawking’s imprimatur.
I found this book so thought-provoking that, on finishing it, I went back to the beginning, so I could re-read his earlier chapters in the context of his later ones. On the very first page he says, rather provocatively, ‘philosophy is dead’. He then spends the rest of the book giving his account of ‘life, the universe and everything’ (which, in one of his early quips, ‘is not 42’). He ends the first chapter (introduction, really) with 3 questions:
1)Why is there something rather than nothing?
2)Why do we exist?
3)Why this particular set of laws and not some other?
It’s hard to get more philosophical than this.
I haven’t read everything he’s written, but I’m familiar with his ideas and achievements, as well as some of his philosophy and personal prejudices. ‘Prejudice’ is a word that is usually used pejoratively, but I use it in the same sense I use it on myself, regarding my ‘pet’ theories or beliefs. For example, one of my prejudices (contrary to accepted philosophical wisdom) is that AI will not achieve consciousness.
Nevertheless, Hawking expresses some ideas that I would not have expected of him. His chapter titled, What is Reality? is where he first challenges the accepted wisdom of the general populace. He argues, rather convincingly, that there are only ‘models of reality’, including the ones we all create inside our heads. He doesn’t say there is no objective reality, but he says that, if we have 2 or more ‘models of reality’ that agree with the evidence, then one cannot say that one is ‘more true’ than another.
For example, he says, ‘although it is not uncommon for people to say that Copernicus proved Ptolemy wrong, that is not true’. He elaborates: ‘one can use either picture as a model of the universe, for our observations of the heavens can be explained by assuming either the earth or the sun is at rest’.
However, as I’ve pointed out in other posts, either the Sun goes around the Earth or the Earth goes around the Sun. It has to be one or the other, so one of those models is wrong.
He argues that we only ‘believe’ there is an ‘objective reality’ because it’s the easiest model to live with. For example, we don’t know whether an object disappears or not when go into another room, nevertheless he cites Hume, ‘who wrote that although we have no rational grounds for believing in an objective reality, we also have no choice but to act as if it’s true’.
I’ve written about this before. It’s a well known conundrum (in philosophy) that you don’t know if you’re a ‘brain-in-a-vat’. But I don’t know of a single philosopher who thinks that they are. The proof is in dreams. We all have dreams that we can’t distinguish from reality until we wake up. Hawking also referenced dreams as an example of a ‘reality’ that doesn’t exist objectively. So dreams are completely solipsistic to the extent that all our senses will play along, including taste.
Considering Hawking’s confessed aversion to philosophy, this is all very Kantian. We can never know the thing-in-itself. Kant even argued that time and space are a priori constructs of the mind. And if we return to the ‘model of reality’ that exists in your mind: if it didn’t accurately reflect the external objective reality outside your mind, the consequences would be fatal. To me, this is evidence that there is an objective reality independent of one’s mind - it can kill you. However, if you die in a dream, you just wake up.
Of course, this all leads to subatomic physics, where the only models of reality are mathematical. But even in this realm, we rely on predictions made by these models to determine if they reflect an objective reality that we can’t see. To return to Kant, the thing-in-itself is dependent on the scale at which we ‘observe’ it. So, at the subatomic scale, our observations may be tracks of particles captured in images, not what we see with the naked eye. The same can be said on the cosmic scale; observations dependent on instruments that may not even be stationed on Earth.
To get a different perspective, I recently read an article on ‘reality’ written by Roger Penrose (New Scientist, 16 May 2020) which was updated from one he wrote in 2006. Penrose has no problem with an ‘objective independent reality’, and he goes to some lengths (with examples) to show the extraordinary agreement between our mathematical models and physical reality.
Our mathematical models of physical reality are far from complete, but they provide us with schemes that model reality with great precision – a precision enormously exceeding that of any description free of mathematics.
(It should be pointed out that Penrose and Hawking won a joint prize in physics for their work in cosmology.)
But Penrose gets to the nub of the issue when he says, ‘...the “reality” that quantum theory seems to be telling us to believe in is so far removed from what we are used to that many quantum theorists would tell us to abandon the very notion of reality’. But then he says in the spirit of an internal dialogue, ‘Where does quantum non-reality leave off and the physical reality that we actually experience begin to take over? Present day quantum theory has no satisfactory answer to this question’. (I try to answer this below.)
Hawking spends an entire chapter on this subject, called Alternative Histories. For me, this was the most revealing chapter in his book. He discusses at length Richard Feynman’s ‘sum over histories’ methodology, called QED or quantum electrodynamics. I say methodology instead of theory, because it’s a mathematical method that has proved extraordinarily accurate in concordance with Penrose’s claim above. Feynman compared it to measuring the distance between New York and Seattle (from memory) to within the width dimension of a human hair.
Basically, as Hawking expounds, in Feynman’s theory, a quantum particle can take every path imaginable (in the famous double-slit experiment, say) and then he adds them altogether, but because they’re waves, most of them cancel each other out. This leads to the principle of superposition, where a particle can be in 2 places or 2 states at once. However, as soon as it’s ‘observed’ or ‘measured’ it becomes one particle in one state. In fact, according to standard quantum theory, it’s possible for a single photon to be split into 2 paths and be ‘observed’ to interfere with itself, as described in this video. (I've edited this after Wes Hansen from Quora challenged it). I've added a couple of Wes's comments in an addendum below. Personally, I believe 'superposition' is part of the QM description of the future, as alluded to by Freeman Dyson (see below). So I don't think superposition really occurs.
Hawking contends that the ‘alternative histories’ inherent in Feynman’s mathematical method, not only affect the future but also the past. What he is implying is that when an observation is made it determines the past as well as the future. He talks about a ‘top down’ history in lieu of a ‘bottom up’ history, which is the traditional way of looking at things. In other words, cosmological history is one of many ‘alternative histories’ (his terminology) that evolve from QM.
This leads to a radically different view of cosmology, and the relation between cause and effect. The histories that contribute to the Feynman sum don’t have an independent existence, but depend on what is being measured. We create history by our observation, rather than history creating us (my emphasis).
As it happens, John Wheeler made the exact same contention, and proposed that it could happen on a cosmic scale when we observed light from a distant quasar being ‘gravitationally lensed’ by an intervening galaxy or black hole (refer Davies paper, linked below). Hawking makes specific reference to Wheeler’s conjecture at the end of his chapter. It should be pointed out that Wheeler was a mentor to Feynman, and Feynman even referenced Wheeler’s influence in his Nobel Prize acceptance speech.
A contemporary champion of Wheeler’s ideas is Paul Davies, and he even dedicates his book, The Goldilocks Enigma, to Wheeler.
Davies wrote a paper which is available on-line, where he describes Wheeler’s idea as the “…participatory universe” in which observers—minds, if you like—are inextricably tied to the concretization of the physical universe emerging from quantum fuzziness over cosmological durations.
In the same paper, Davies references and attaches an essay by Freeman Dyson, where he says, “Dyson concludes that a quantum description cannot be applied to past events.”
And this leads me back to Penrose’s question: how do we get the ‘reality’ we are familiar with from the mathematically modelled quantum world that strains our credulousness? If Dyson is correct, and the past can only be described by classical physics then QM only describes the future. So how does one reconcile this with Hawking’s alternative histories?
I’ve argued elsewhere that thepath from theinfinitely many paths of Feynman’s theory, is only revealed when an ‘observation’ is made, which is consistent with Hawking’s point, quoted above. But it’s worth quoting Dyson, as well, because Dyson argues that the observer is not the trigger.
... 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.
But, as I’ve pointed out in other posts, consciousness exists in a constant present. The time for ‘us’ is always ‘now’, so the ‘point of reference’, that is key to Dyson’s argument, correlates with the ‘now’ of a conscious observer.
We know that ‘decoherence’ is not necessarily dependent on an observer, but dependent on the wave function interacting with ‘classical physics’ objects, like a laboratory apparatus or any ‘macro’ object. Dyson’s distinction between past and future makes sense in this context. Having said that, the interaction could still determine the ‘history’ of the quantum event (like a photon), even it traversed the entire Universe, as in the cosmic background radiation (for example).
In Hawking’s subsequent chapters, including one titled, Choosing Our Universe, he invokes the anthropic principle. In fact, there are 2 anthropic principles called the ‘weak’ and the ‘strong’. As Hawking points out, the weak anthropic principle is trivial, because, as I’ve pointed out, it’s a tautology: Only universes that produce observers can be observed.
On the other hand, the strong anthropic principle (which Hawking invokes) effectively says, Only universes that produce observers can ‘exist’. One can see that this is consistent with Davies’ ‘participatory universe’.
Hawking doesn’t say anything about a ‘participatory universe’, but goes into some detail about the fine-tuning of our universe for life, in particular the ‘miracle’ of how carbon can exist (predicted by Fred Hoyle). There are many such ‘flukes’ in our universe, including the cosmological constant, which Hawking also discusses at some length.
Hawking also explains how an entire universe could come into being out of ‘nothing’ because the ‘negative’ gravitational energy cancels all the ‘positive’ matter and radiation energy that we observe (I assume this also includes dark energy and dark matter). Dark energy is really the cosmological constant. Its effect increases with the age of the Universe, because, as the Universe expands, gravitational attraction over cosmological distances decreases while ‘dark energy’ (which repulses) doesn’t. Dark matter explains the stable rotation of galaxies, without which, they’d fly apart.
Hawking also describes the Hartle-Hawking model of cosmology (without mentioning James Hartle) whereby he argues that in a QM only universe (at its birth), time was actually a 4th spatial dimension. He calls this the ‘no-boundary’ universe, because, as John Barrow once quipped, ‘Once upon a time, there was no time’. I admit that this ‘model’ appeals to me, because in quantum cosmology, time disappears mathematically.
Hawking’s philosophical view is the orthodox one that, if there is a multiverse, then the anthropic principle (weak or strong) ensures that there must be a universe where we can exist. I think there are very good arguments for the multiverse (the cosmological variety, not the QM multiple worlds variety) but I have a prejudice against an infinity of them because then there would be an infinity of me.
Hawking is a well known atheist, so, not surprisingly, he provides good arguments against the God hypothesis. There could be a demiurge, but if there is, there is no reason to believe it coincides with any of the Gods of mythology. Every God I know of has cultural ties and that includes the Abrahamic God.
For someone who claims that ‘philosophy is dead’, Hawking’s book is surprisingly philosophical and thought-provoking, as all good philosophy should be. In his conclusions, he argues strongly for ‘M theory’, believing it will provide the theory(s) of everything that physicists strive for. M theory, as Hawking acknowledges, requires ‘supersymmetry’, and from what I know and read, there is little or no evidence of it thus far. But I agree with Socrates that every mystery resolved only uncovers more mysteries, which history, thus far, has confirmed over and over.
My views have evolved and, along with the ‘strong anthropic principle’, I’m becoming increasingly attracted to Wheeler’s ‘participatory universe’, because the more of its secrets we learn, the more it appears as if ‘the Universe saw us coming’, to paraphrase Freeman Dyson.
Addendum (23Apr2021): Wes Hansen, whom I met on Quora, and who has strong views on this topic, told me outright that he's not a fan of Hawking or Feynman. Not surprisingly, he challenged some of my views and I'm not in a position to say if he's right or wrong. Here are some of his comments:
You know, I would add, the problem with the whole “we create history by observation” thing is, it takes a whole lot of history for light to travel to us from distant galaxies, so it leads to a logical fallacy. Consider:
Suppose we create the past with our observations, then prior to observation the galaxies in the Hubble Deep Fields did not exist. Then where does the light come from? You see, we are actually seeing those galaxies as they existed long ago, some over 10 billion years ago.
Regarding his last point, I think Ian Miller has a point. I don't always agree with Miller, but he has more knowledge on this topic than me. I argue that the superposition, which we infer from the interference pattern, is in the future. The idea of a single photon taking 2 paths and interfering with itself is deduced solely from the interference pattern (see linked video in main text). My view is that superposition doesn't really happen - it's part of the QM description of the future. I admit that I effectively contradicted myself, and I've made an edit to the original post to correct that.
I’ve said this before, but it’s worth repeating: no one totally agrees with everything by someone else. In fact, we each of us change our views as we learn and progress and become exposed to new ideas. It’s okay to cherry-pick. In fact, it’s normal. All the giants in science and mathematics and literature and philosophy borrowed and built on the giants who went before them.
I’ve been reading about Plato in A.C. Grayling’s extensive chapter on him and his monumental status in Western philosophy (The History of Philosophy). According to Grayling, Plato was critical of his own ideas. His later writings challenged some of the tenets of his earlier writings. Plato is a seminal figure in Western culture; his famous Academy ran for almost 800 years, before the Christian Roman Emperor, Justininian, closed it down in 529 CE, because he considered it pagan. One must remember that it was during the Roman occupation of Alexandria in 414 that Hypatia was killed by a Christian mob, which many believe foreshadowed the so-called ‘Dark Ages’.
Hypatia had good relations with the Roman Prefect of her time, and even had correspondence with a Bishop (Synesius of Cyrene), who clearly respected, even adored her, as her former student. I’ve read the transcript of some of his letters, care of Michael Deakin’s scholarly biography. Deakin is Honorary Research Fellow at the School of Mathematical Sciences of Monash University (Melbourne, Australia). Hypatia also taught a Neo-Platonist philosophy, including the works of Euclid, a former Librarian of Alexandria. On the other hand, the Bishop who is historically held responsible for her death (Cyril) was canonised. It’s generally believed that her death was a ‘surrogate’ attack on the Prefect.
Returning to my theme, the Academy of course changed and evolved under various leaders, which led to what’s called Neoplatonism. It’s worth noting that Augustine was influenced by Neoplatonism as well as Aquinas, because Plato’s perfect world of ‘forms’ and his belief in an immaterial soul lend themselves to Christian concepts of Heaven and life after death.
But I would argue that the unique Western tradition that combines science, mathematics and epistemology into a unifying discipline called physics has its origins in Plato’s Academy. It was a pre-requisite, specified by Plato, that people entering the Academy required a knowledge of mathematics. The one remnant of Plato’s philosophy, which stubbornly resists being relegated to history as an anachronism, is mathematical Platonism, though it probably means something different to Plato’s original concept of ‘forms’.
In modern parlance, mathematical Platonism means that mathematics has an independent existence to the human mind and even the Universe. To quote Richard Feynman (who wasn’t a Platonist) from his book, The Character of Physical Law in the chapter titled The Relation of Mathematics to Physics.
...what turns out to be true is that the more we investigate, the more laws we find, and the deeper we penetrate nature, the more this disease persists. Every one of our laws is a purely mathematical statement in rather complex and abstruse mathematics... Why? I have not the slightest idea. It is only my purpose to tell you about this fact.
The ’disease’ he’s referring to and the ‘fact’ he can’t explain is best expressed in his own words:
The strange thing about physics is that for the fundamental laws we still need mathematics.
To put this into context, he argues that when you take a physical phenomenon that you describe mathematically, like the collision between billiard balls, the fundaments are not numbers or formulae but the actual billiard balls themselves (my mundane example, not his). But when it comes to fundaments of fundamental laws, like the wave function in Schrodinger’s equation (again, my example), the fundaments remain mathematical and not physical objects per se.
In his conclusion, towards the end of a lengthy chapter, he says:
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.
I’m not aware of any physicist who would disagree with that last statement, but there is strong disagreement whether mathematical language is simply the only language to describe nature, or it’s somehow intrinsic to nature. Mathematical Platonism is unequivocally the latter.
Grayling’s account of Plato says almost nothing about the mathematical and science aspect of his legacy. On the other hand, he contends that Plato formulated and attempted to address three pertinent questions:
What is the right kind of life, and the best kind of society? What is knowledge and how do we get it? What is the fundamental nature of reality?
In the next paragraph he puts these questions into perspective for Western culture.
Almost the whole of philosophy consists in approaches to the related set of questions addressed by Plato.
Grayling argues that the questions need to be addressed in reverse order. To some extent, I’ve already addressed the last two. Knowledge of the natural world has become increasingly dependent on a knowledge of mathematics. Grayling doesn’t mention that Plato based his Academy on Pythagoras’s quadrivium: arithmetic, geometry, astronomy and music; after Plato deliberately sought out Pythagoras’s best student, Archytas of Terentum. Pythagoras is remembered for contending that ‘all is number’, though his ideas were more religiously motivated than scientific.
But the first question is the one that was taken up by subsequent philosophers, including his most famous student, Aristotle, who arguably had a greater and longer lasting influence on Western thought than his teacher. But Aristotle is a whole other chapter in Grayling’s book, as you’d expect, so I’ll stick to Plato.
Plato argued for an ‘aristocracy’ government run by a ‘philosopher-king’, but based on a meritocracy rather than hereditary rulership. In fact, if one goes into details, he effectively argued for leadership on a eugenics basis, where prospective leaders were selected from early childhood and educated to rule.
Plato was famously critical of democracy (in his time) because it was instrumental in the execution of his friend and mentor, Socrates. Plato predicted that democracy led to either anarchy or the rule of the wealthy over the poor. In the case of anarchy, a strongman would logically take over and you'd have 'tyranny', which is the worst form of government (according to Plato). The former (anarchy) is what we’ve recently witnessed in so-called 'Arab spring' uprisings.
The latter (rule by the wealthy) is what has arguably occurred in America, where lobbying by corporate interests increasingly shapes policies. This is happening in other ‘democracies’, including Australia. To give an example, our so-called ‘water policy’ is driven by prioritising the sale of ‘water rights’ to overseas investors over ecological and community needs; despite Australia being the driest continent in the world (after Antarctica). Keeping people employed is the mantra of all parties. In other words, as long as the populace is gainfully employed, earning money and servicing the economy, policy deliberations don’t need to take them into account.
As Clive James once pointed out, democracy is the exception, not the norm. Democracies in the modern world have evolved from a feudalistic model, predominant in Europe up to the industrial revolution, when social engineering ideologies like fascism and communism took over from monarchism. It arguably took 2 world wars before we gave up traditional colonial exploitation, and now we have exploitation of a different kind, which is run by corporations rather than nations.
I acknowledge that democracy is the best model for government that we have, but those of us lucky enough to live in one tend to take if for granted. In Athens, in the original democracy (in Plato’s time) which was only open to males and excluded slaves, there was a broad separation between the aristocracy and the people who provided all the goods and services, including the army. One can see parallels to today’s world, where the aristocracy have been replaced by corporate leaders, and the interdependence and political friction between these broad categories remain. In the Athens Senate (according to historian, Philip Matyszak) if you weren’t an expert in the field you pontificated on, like ship building (his example) you were generally given short thrift by the Assembly.
I sometimes think that this is the missing link in today’s governance, which has been further eroded by social media. There are experts in today’s world on topics like climate change and species extinction and water conservation (to provide a parochial example) but they are often ignored or sidelined or censored. As recently as a couple of decades ago, scientists at CSIRO (Australia’s internationally renowned, scientific research organisation) were censored from talking about climate change, because they were bound by their conditions of employment not to publicly comment on political issues. And climate change was deemed a political issue, not a scientific one, by the then Cabinet, who were predominantly climate change deniers (including the incumbent PM).
In contrast, the recent bush fire crisis and the current COVID-19 crisis have seen government bodies, at both the Federal and State level, defer to expertise in their relevant fields. To return to my opening paragraph, I think we can cherry-pick some of Plato’s ideas in the context of a modern democracy. I would like to see governments focus more on expertise and long-term planning beyond a government’s term in office. We can’t have ‘philosopher kings’, but we do have ‘elite’ research institutions that can work with private industries in creating more eco-friendly policies that aren’t necessarily governed by the sole criterion of increasing GDP in the short term. I would like to see more bipartisanship rather than a reflex opposition to every idea that is proposed, irrespective of its merits.
I recently wrote a comment on Quora that addresses this very question, but I need to backtrack a couple of decades. When I studied philosophy, I wrote an essay on Kant, around the same time I wrote my essay on Christianity and Buddhism.
Not so long ago (over Christmas) I read AC Grayling’s The History of Philosophy, which, at 580+ pages is pretty extensive and even includes brief discussions on Hindu, Chinese, Islamic and sub-Saharan African philosophy. Any treatise you read on the history of Western philosophy will include Kant as one of the giants of the discipline. Grayling’s book, in particular, provides both historical and contextual perspectives. According to Grayling, Kant brought together the two ‘opposing’ branches of analytical philosophy of his time: empiricism and idealism.
I’ve read Critique of Pure Reason (in English, obviously) and it’s as obscure in places as Kant’s reputation presumes. Someone once claimed that Kant’s lectures were very popular and a lot less intimidating than his texts. If that is true, then one regrets that he didn’t live in the age of YouTube. But his texts, and subsequent commentaries on them, are all we have, including this one you’re about to read. I will include the original bibliography, as I did with my other ‘academic’ essay.
The essay was titled: What is transcendental idealism?
Kant, I believe, made two major contributions to philosophy: that there is a limit to what we can know; and that there is a difference between what we perceive and ‘things-in-themselves’. These two ideas are naturally related but they are not synonymous. Transcendental idealism arose out of Kant’s attempt to incorporate these ideas into an overall philosophy of knowledge or epistemology. Kant is extremely difficult to follow and this is not helped when many of the essays written on Kant are just as obtuse and difficult to understand as Kant himself. However there are parts of Kant’s Critique that are relatively plain and easy to follow. It is my intention to start with these aspects and work towards an exposition on transcendental idealism.
I think it is important to note that our understanding in science and psychology has increased considerably since Kant’s time, and this must influence any modern analysis of his epistemology. For example, in Kant’s time, it was Newton’s physics that provided the paradigm for empirical knowledge and therefore a deterministic universe seemed inevitable. With the discovery of quantum mechanics and Chaos theory, this is no longer the case, and Kant’s third 'antimony' on ‘freedom’ does not have the same relevance as it did in his time. A contemporary analogy to this might be materialism as the current paradigm for consciousness, because current theories are based on our knowledge of genetics, biochemistry and neuroscience, and the limitations of that knowledge. It is quite possible that future developments may overturn materialism as a paradigm because our knowledge of consciousness today is arguably no greater than our knowledge of physics was during Newton’s time.
In view of what we’ve learnt since Kant’s time, it seems to me that he had a remarkable, indeed almost prophetic insight, yet I cannot help but also believe that his philosophy contains a fundamental flaw. The fundamental flaw is his insistence that space and time are purely psychological phenomena, or in Kant’s own terms, that space and time are apriori ‘forms’ of the mind. ‘But this space and this time, and with them all appearances, are not in themselves things; they are nothing but representations and cannot exist outside our minds.’ One of my objectives, therefore, is to provide a resolution of this flaw with aspects of his philosophy that I find sound. Ironically, I believe that time and space give us the best insight into understanding Kant’s transcendental idealism, though not in a manner that he could have foreseen.
A philosophy of knowledge naturally includes knowledge acquisition, and for Kant, this required an analysis of human cognitive abilities. I believe this is a good place to start in understanding Kant. Kant realised that there are two aspects of knowledge acquisition in humans: what we gain directly through our senses or ‘sensibilities’ and what we ‘synthesise’ into concepts through ‘pure understanding’. Kant realised that this synthesis is in effect consciousness. Kant explains how concepts can go beyond experience, which is what he calls pure understanding. This in effect is transcendental idealism, which is speculative as opposed to empirical realism which is based on experience. Another perspective to this is that most animals, we assume, can synthesise knowledge at the sensibility level, otherwise they would not be able to interact with their environment, whereas humans can synthesise knowledge at another level altogether which I believe is Kant’s transcendental level. Note that Kant is not talking about metaphysical knowledge in his reference to the transcendental, but knowledge of the object-in-itself, a concept I will return to later.
Whether Kant realised it or not, this synthesis of concepts is also the way in which we remember things in the long term - that is through association of concepts. I’m talking about knowledge type memory rather than physiological type memory which allows us to remember how to do tasks, like driving a car or playing a musical instrument. These are different types of memory which are dependent on different physiological mechanisms within the brain. The point is that this synthesising of concepts is a memory function as well as a means of understanding. It is virtually impossible to remember new knowledge unless we synthesise it into existing knowledge.
Both in the Study Guide and in Allison’s essay on The Thing in Itself, perception of colour is used as an example of knowledge gained through the senses, and in the Study Guide is contrasted with space and time, which according to Kant are apriori knowledge, and therefore independent of experience. This leads to the problem I have with Kant, because space and time are also sensed by us, despite Kant’s objections that space and time are not entities. It should be pointed out that colour is purely a psychological phenomenon. In other words, colour, unlike space and time, does not exist outside the mind. In fact colour is probably the best example for explaining the difference between what we perceive (our ‘representations’) and ‘things-in-themselves’. Colour as it is-in-itself is a wavelength of light, and so is radar and radio waves and cosmic rays. It is believed that some animals can see in ultraviolet light so that for them ultraviolet light is a colour. Colour best explains Kant’s philosophical point that appearances or representations are not the same as the phenomenon as it exists-in-itself.
So colour only exists in the mind as the result of sensory perception, as Kant himself explained. It is not that appearances or representations of objects as perceived are different entities to what exists in the real world, but that we are only aware or can only sense specific attributes of these objects. This is an important point that is not often delineated.
So in what respects are space and time different? Space and time are different because they are the manifold in which the universe exists - without space and time there would be no universe, no physical universe anyway; no universe that we could perceive in an empirical sense, therefore no empirical realism. According to Kant however, space and time are apriori ‘forms’ that we impose on the universe. There are many aspects to this issue so let’s start with sensory perception. In regard to space, we have a sense in addition to the five known ones called proprioception, discovered by Sherrington in the 1890s. This is a sense that tells us where every part of our body is in space. Oliver Sacks in his book, The Man Who Mistook His Wife for a Hat, describes a case he called ‘The Disembodied Lady’, of a woman who lost this sense completely overnight. She was literally like a rag doll and had to learn to do even the most simple motor tasks, like sitting, anew. But of course we also sense space with our eyes, and all animals that depend on their dynamic abilities, from insects to birds, to mammals, have this ability. Bats and dolphins of course sense space with echo-location.
As for time, we have two means of sensing time. The most obvious is memory. Again Sacks describes the case of a man suffering retrograde amnesia, which in his book he called ‘The Lost Mariner’. Sacks met the man in 1975, but although he displayed above average intelligence, the man could create no new memories. In fact he was permanently stuck in 1945 when he had left the US Navy after the War. This is like being colour blind or deaf beyond a certain frequency. The other sense of time is through our eyes which capture images at a very specific rate. Without this ability we would not be able to detect motion. All photographs, to use an analogy, need time, no matter how small an increment, in order to be realised at all. Again different animals capture these images at different rates so they quite literally live at different speeds. Birds and many insects see the world in slow motion compared to us, whereas other animals like snails and sloths see it much faster. Sometimes in the event of trauma, like a car accident or an explosion, our internal clock changes its rate momentarily and we see things as if we are watching a slow motion film.
We sense space and time the same way we sense colours, sounds and smells. In fact our ability to sense space and time is a matter of life and death - just take a drive in traffic. The idea that we impose space and time on the universe is absurd unless one believes in solipsism which apparently Kant did not. For Kant time and space are apriori knowledge that is ‘given’. Our mind has an inbuilt sense of time and space, yes, but it is a necessary sense no different to our other senses so that we can interact with a world that exists in time and space. This is the distinction I make with Kant. The reason we have a sense of space and time is so the world inside our heads can match the world outside our heads, otherwise we could not do anything - we could not even walk outside our front doors. To argue otherwise, in my opinion, is disingenuous.
This contention on my part has consequences for Kant’s philosophy. According to the Encyclopaedia Britannica, Kant’s ‘Copernican revolution of philosophy’ is: ‘...the assumption not that man’s knowledge must conform to objects but that objects must conform to man’s apparatus of knowing.’ I would turn this argument on its head because it is my belief that the human mind is a mirror of the physical world and not the other way round. Michio Kaku and Jennifer Thompson in their book, Beyond Einstein, describe the hypothetical experience of meeting someone from a higher dimensional universe. They explain that whilst we can perceive things in 2 dimensions of space, if we lived in a 2 dimensional space, 3 dimensions would be incomprehensible to us. If we lived in a higher dimensional universe we would think in those higher dimensions. This is why we can’t create a higher dimensional universe in our imaginations but we can express it mathematically. This I believe also gives us an insight into transcendental idealism, but I will return to this point later.
In terms of our sensibilities, Kant is correct: our ability to perceive is limited by the cognitive powers of the human mind. We cannot see colours outside a certain range of wavelength of light or hear sounds outside a specific range of frequencies. But Kant goes further than this: he realised that our cognitive reasoning ability to understand the things-in-themselves is also limited. Kant quite correctly realised that there is a trap or an illusion, that we often perceive concepts which we synthesise through our reasoning ability as being derived from experience when they are not. We have these ideas in our head which we believe to match reality, but in truth we only think we understand reality and the thing-in-itself escapes us. This is the kernel in the midst of Kant’s philosophy which is worth preserving. Our knowledge acquisition is in fact an interaction between experience (the empirical) and theory (the transcendental). Kant himself showed an insight into this interaction in A95 when he refers to the synthesis of ‘sense, imagination and apperception’.
‘All these faculties have a transcendental (as well as an empirical) employment which concerns the form alone, and is possible apriori.’ By ‘apriori’ and ‘form’, Kant of course is referring to space and time, but he is also referring to mathematical forms, as he explains on the next page in B128. There is then, this relationship between transcendental idealism and empirical realism; a relationship that is mediated principally through mathematics.
But there is another aspect of our knowledge acquisition that Kant never touched on and relates to the thing-in-itself. We have discovered that nature takes on completely different realities at different levels which means that the thing-in-itself is almost indefinable as a single entity. To describe something we have to conceptually isolate it in our minds. For example the human body is a single entity made up of millions of other entities called cells. It is virtually impossible to conceptualise these two levels of entities simultaneously. But the human mind has a very unique ability. We can create concepts within concepts, like words within sentences, or formulas within mathematical equations, or notes within music, and realise that on different levels all these things take on different meanings. So the human mind is uniquely placed to understand the universe in which we live, because it also takes on different meanings at different levels.
This is even true regarding the number of dimensions of the universe. Michio Kaku, whom I referred to earlier, informs us that according to M theory, the universe may very well exist at one level in 11 dimensions, but at our level of everyday existence, we can only perceive the 3 dimensions of space and the 1 dimension of time. This for me is the irony of Kant’s philosophy. Relativity theory and quantum mechanics suggest that space and time are not how we perceive them to be, which makes Kant’s concept of the thing-in-itself quite a prophetic insight. However Kant would never have conceived that space and time could exist as things-in-themselves at all, because for Kant, space and time are not entities. He is right in that they are not entities in the same way that objects are, but they are the absolutely essential components for the universe to exist at all.
Some people would argue that space and time are no more than mathematical entities, because that is the only way we can express space and time, as opposed to how we experience it. From this argument it could be suggested that by using mathematics we are imposing our sense of space and time on the universe, irrespective of all the arguments I have already made concerning how we are able to sense it. But what I find significant is that mathematical laws are not man made and that nature obeys them even if we weren’t here to express them. So I would argue that transcendental idealism is mathematics, even though I’m not at all sure if Kant would concur. I think Pythagoras showed remarkable insight when he claimed that all things are numbers, even though he was talking from a religious perspective. But metaphysics aside, Pythagoras was one of the first philosophers to understand that mathematics gives us a rare and unique insight into the natural world. What would he think today? What’s more I think Pythagoras would be quite agreeable in thinking that Kant’s transcendental idealism was indeed the world of mathematics.
Bibliography
Kaku M., Hyperspace, Oxford University Press, 1994.
Kaku M. & Thompson J., Beyond Einstein, Oxford University Press, 1997.
Kant I., Smith N. (trans.), Critique of Pure Reason, Macmillan, London, 1929.
Philosophy, The History of Western, Encyclopaedia Britannica, Vol.25, Edition 15, 1989, pp.742-69.
Reason And Experience, Theories of Knowledge B, Reader, Deakin University, Geelong, Australia, 1989.
Reason And Experience, Theories of Knowledge B, Study Guide, Deakin University, Geelong, Australia, 1989.
Ross K., Immanuel Kant, web page http://www.friesian.com/kant.htm
Sacks, O., The Man Who Mistook His Wife for a Hat, Picador, London, 1986.
Sternberg R., In Search of the Human Mind, Yale University, Harcourt Brace College Publishers, 1995.
