How algebra turned mathematics into a language

A little while ago I wrote a post arguing that mathematics as language was just a metaphor. I’ve since taken the post down, though those who subscribe may still have a copy. In the almost 10 years I’ve been writing this blog it’s only the second time I’ve deleted a post. The other occasion was very early in its life when I posted an essay on existentialism (from memory) only to post something more relevant.

The reason I took the post down was because I thought I was being a bit petty in criticising some guy on YouTube who was probably actually doing some good in the world, even if I disagreed with him on a philosophical level. Instead, I wrote a comment on his video, challenging the premise of his talk that the reason mathematics is ‘difficult’ for many people is because it’s not taught as a language. I would still challenge the validity of that premise, but I would now change my own approach by acknowledging that there is a sense in which mathematics is a language, but not in a lingua franca sense.

In my last post – the review of Arrival – language and communication are major themes, and I make mention of a piece of expositional dialogue that I thought very insightful and stuck in my brain as a revelatory thought. To remind everyone: it was the realisation that language determines the limits of what we can think because we all think in a language. In other words, if a language doesn’t define the specific concepts we are trying to comprehend then we struggle to conjure up those concepts, and mathematics provides a good example.

The reason that mathematics is best not construed as a language is because mathematics, as it’s generally practiced, has its own language and that language is algebra. As I’ve said before: mathematics is not so much about numbers as the relationship between numbers, and the efficacy of algebra is that it allows one to see the relationships without the numbers.

And this is the thing, because some people find it easier to think in algebra than others. I will illustrate with examples.

A = k/B then B = k/A

If k is a constant (can’t change) and A and B are variables then there is an inverse relationship between A and B. In other words, if A gets larger then B must get smaller and vice versa. This can be written as A α 1/B or B α 1/A, where α (in this context) means ‘is proportional to’. Note that if the number on the bottom gets smaller then the whole term must get larger and, of course, the converse is also true: if the number on the bottom gets larger then the whole term must get smaller.

People who are familiar with these concepts think this automatically. They also know that if you move a term from one side of an equation to the other, then you either invert it or take its negative. So if you have a language that captures these concepts, then you can think in these concepts with no great effort. It also means that you are not easily intimidated by equations.

To give another common example: the distributive rule, which is arguably the most commonly used rule in algebra.

A = B(C + D) is the same as A = BC + BD

And if A = -B(C - D) then A = BD – BC

(Note that multiplying by minus changes the sign: from + to - and - to +)

We could have done this differently because –(C – D) = D – C and B(D – C) = BD –BC   (So same answer)

This is all very simple stuff and it can be extended to include square roots (including square roots of -1), logarithms, trig functions and so on. Even calculus is just algebra with numbers disappearing into zero with the inverse of infinity (called infinitesimals).

One of the problems in learning mathematics is that we are trying to learn new concepts and simultaneously a new ‘language’ of symbols. But if the language of algebra allows one to think in new concepts, then a hurdle becomes a springboard to new knowledge.

Arrival; a masterclass in storytelling

Four movie reviews in one year; maybe I should change the title of my blog – no, just kidding. Someone (either Jake Wilson or Paul Byrnes from The Age) gave it the ultimate accolade: ‘At last, a science fiction movie with a brain.’ They also gave it 3.5 stars but ended their review with: ‘[the leads: Amy Adams, Forest Whitaker and Jeremy Renner] have the chops to keep us watching even when the narrative starts to wobble.’ So they probably wouldn’t agree with me calling it a masterclass.

It’s certainly not perfect – I’m not sure I’ve seen the perfect movie yet – but it’s clever on more than one level. I’m always drawn to good writing in a movie, which is something most people are not even aware of. It was based on a book, whose author escaped me as a couple in front of me got up to leave just as the name came up on the screen. But I have Google, so I can tell you that the screenplay was written by Eric Heisserer, and Ted Chiang wrote the novella, “Story of Your Life”, upon which it is based. French-Canadian director, Denis Villeneuve has also made Prisoners and Sicario, neither of which I’ve seen, but Sicario is highly acclaimed.

It would be remiss of me not to mention the music and soundscape, which really adds another dimension to this movie. I noticed that beginning and end scores were by Max Richter, whom I admire in the contemporary classical music scene. Though the overall music score is credited to Johann Johannsson. Some of the music reminded of Tibetan music with its almost subterranean tones. Australia also gets a bit of 'coverage', if that's the right word, though not always in a flattering manner. Forest Whitaker's character reminds us how we all but committed genocide against the Aboriginal people.

I haven’t read the book, but I’m willing to give credit to both writers for producing a ‘science fiction story with a brain’. Science fiction has a number of subgenres: the human diaspora into interstellar space; time travel; alien worlds; parallel universes; artificial intelligence; dystopian fiction, utopian fiction and the list goes on, with various combinations. The title alone tells us that this is an Alien encounter on Earth, but the movie keeps us guessing as to whether it’s an invasion or just a curious interloper or something else altogether.

I’ve written elsewhere that narrative tension is one of the essential writing skills and this story has it on many levels. To give one example without giving the plot away, there is a sequence of narrative events where we think we know what’s going to happen, with the suspense ramping up while we wait for what we expect to happen to happen, then something completely unexpected happens, which is totally within the bounds of possibility, therefore believable. In some respects this sums up the whole movie because all through it we are led to believe one thing only to learn we are witnessing something else. It’s called a reversal, which I’m not always a fan of, but this one is more than just a clever twist for the sake of being clever. Maybe that’s what the reviewer meant by ‘…when the narrative starts to wobble’. I don’t know. I have to confess I wasn’t completely sold, yet it was essential to the story and it works within the context of the story, so it’s part of the masterclass.

One of the things that struck me right from the beginning is that we see the movie almost in first person – though, not totally, as at least one cutaway scene requires the absence of the protagonist. I would not be surprised if Ted Chiang wrote his short story in the first person. I don’t know what nationality Ted Chiang is, but I assume he is of Chinese extraction, and the Chinese are major players in this movie.

Communication is at the core of this film, both plot and subplot, and Amy Adams’ character (Louise Banks) makes the pertinent point in a bit of expositional dialogue that was both relevant to the story and relevant to what makes us human: that language, to a large extent, determines how we think because, by the very nature of our brains, we are limited in what we can think by the language that we think in. That’s not what she said but that was the lesson I took from it.

I’ve made the point before, though possibly not on this blog, that science fiction invariably has something to say about the era in which it was written and this movie is no exception. Basically, we see how paranoia can be a dangerous contagion, as if we need reminding. We are also reminded how wars and conflicts bring out the best and worst in humanity with the worst often being the predominant player.

When evolution is not evolution

No, I’m not talking about creationism (a subject I’ve discussed many times on this blog) but a rather esoteric argument produced by Donald D Hoffman and Chetan Prakash in an academic paper titled Objects of Consciousness. Their discussion on evolution is almost a side issue, and came up in their responses to the many objections they’ve fielded. I read the paper when I was sent a link by someone who knows I’m interested in this stuff.

Donald Hoffman is a cognitive scientist with a Ph.D. in Computational Psychology and is now a full professor at University of California, Irvine. Chetan Prakash is a Professor Emeritus at California State University, San Bernardino and has a Master of Science in Physics and a Master of Science in Applied Mathematics.

I should point out at the outset, that their thesis is so out there, that I seriously wondered if it was a hoax. But given their academic credentials and the many academic citations and references in their paper, I assume that the authors really believe in what they’re arguing. And what they’re arguing, in a nutshell, is that everyone’s (and I mean every person’s) perception of the world is false, because, aside from conscious agents, everything else, including spacetime, is impermanent.

Their paper is 20 pages long (including 5-6 pages of objections and replies) most of which are densely worded interspersed with some diagrams and equations. To distil someone’s treatise into a single paragraph is always a tad unfair, so I’ll rely heavily on direct quotations and references to impart their arguments. Besides, you can always read the entire paper for yourself. Basically, they argue that ‘interacting conscious agents’ are the only reality and that nothing else exists ‘unperceived’. They formulate a mathematical model of consciousness, from which they derive a wave function that is the bedrock of quantum mechanics (which I’ll refer to as QM for brevity). In other words, they argue that the Copenhagen interpretation of QM requires consciousness to bring objects into reality (except consciousness) which are all impermanent.

It’s a well known philosophical conundrum that you can’t prove that you’re not a ‘brain-in-a-vat’, and theirs is a similar point of view in that it can’t be proved that they’re wrong, even though, as they point out themselves, we mostly all believe their view is wrong. I don’t know of anyone (other than the authors) who think that the world ceases to exist when they’re not looking. This is known as solipsism and there is a very good argument against solipsism even though it can’t be proved it’s wrong. In fact, solipsism is absolutely true when you’re in a dream, so it’s not always wrong. The point is that when we’re in a dream, despite all its inconsistencies, we actually don’t know we’re in a dream, so how can you be sure you’re not in a dream when you’re consciously awake? The argument against solipsism is that it can only be held by one person: it’s impossible to believe that everyone else is a solipsist too.

In the objections, item 6, they ‘reject solipsism’, yet ‘also reject permanence, viz., the doctrine that 3D space and physical objects exist when they are not perceived [but not conscious agents]. To claim that conscious agents exist unperceived differs from the claim that unconscious objects and space-time exist unperceived.’ In other words, consciousness is the only reality, a point they make in response to Objection 19: ‘reality consists of interacting conscious agents.’ But if one takes this seriously, then even the bodies that we take for granted don’t exist ‘unperceived’ whilst our consciousness does. It’s utter nonsense, except in a dream. What they are describing is exactly the reality one perceives in a dream, so their theory is effectively that the reality we all believe we inhabit is, in effect, a dream. Which is logically a variation on solipsism. The only difference is that we all inhabit the same dream together. So we’re all brains in a vat, only connected. The authors, I’m sure, would reject this interpretation, yet it fits exactly with what they’re arguing. Only in a dream do objects, including our own bodies, cease to exist unperceived.

Evolution comes up a lot in their paper because one of the centrepieces of their thesis is that evolution by natural selection produces perceptions that favour ‘fitness’ over ‘truth’. They claim to run 'genetic algorithms’ that show that evolution by natural selection benefits perception for ‘fitness’ over ‘accuracy’. The point is that we must take this assertion on face value, because we don’t know what algorithms they’re using or how they even define fitness, perceptions and truth. In fact, Objection 12 asks this very question. Part of the authors' response goes: ‘For the sake of brevity, we omitted our definition of truth and perception… But they are defined precisely in Monte Carlo simulations of evolutionary games and genetic algorithms…’

In particular, the authors use vision to make their case. It’s well known that the brain creates a facsimile of what we see in ways that we are still trying to understand, and to which, to date, we’ve failed to engineer to the same degree of accuracy in artificial intelligence (AI). But theoretical algorithms and Monte Carlo simulations aside, we have the means to compare what we subjectively see with an objective representation.

It so happens that we have invented devices that create images (both stationary and dynamic) through chemical-electronic-mechanical means independently of the human brain and they show remarkable, but unsurprising, veracity with what our brain perceives subjectively. Now, you might say that the same brain perceives this simulated vision, so one would expect it to provide the same image. I think this is a long bow to draw, because the image effectively gets ‘processed’ twice: once through the device and once through the brain, yet the result is unequivocally the same without the interim process. In fact, the interim process can show what we miss, like the famous example of a gorilla moving through a room while you are concentrating on a thrown ball. But, in the context of their thesis, the camera is not a conscious entity yet it captures an image that is supposedly nonexistent when unperceived. And cameras can be set up to capture images without the interaction of so-called ‘conscious agents’.

Now the authors are correct when they point out that colour, for example, is a completely psychological phenomenon – it only exists in some creature’s mind, and it varies from species to species – this is well known and well understood. We also know that it’s caused by reflected light which can be scientifically explained by Richard Feynman’s (I know it’s not his alone) QED (Quantum Electrodynamics) and that the subjective experience of colour is a direct consequence of the frequency of electromagnetic radiation.  But the fact that colour is subjective doesn’t make the objects, from which the effect is consequential, subjective as well.

Regarding the other mathematical contribution to their thesis, the authors have created a mathematical model of consciousness, from which they derive the wave function for QM. I’m not a logician, so I can’t say one way or another how valid this is. However, it should be pointed out that Erwin Schrodinger, who originally proposed the wave function, in his famous eponymous equation, didn’t derive it from anything. So the authors claim they’ve done something that the original creator of the wave function couldn’t do himself. As Richard Feynman once said: ‘Schrodinger’s equation can’t be derived from anything we know.’ However, the authors claim it can be derived from consciousness. I’m sceptical.

You may wonder what all this has to do with the title of this post. Well, in response to objection 19, the authors propose to come up with a ‘new theory of evolution’ based on their theory of conscious agents. To quote: ‘When the new evolutionary theory is projected onto the spacetime perceptual interface of H. Sapiens we must get back the standard evolutionary theory.’ This means that the DNA, and the molecules that make the DNA, that allowed consciousness to evolve are actually dependent on said consciousness, so the ‘new theory of evolution’ must logically contradict the ‘standard theory of evolution’.

As part of their thesis, the authors make an analogy between a computer desktop and spacetime, only, the way they describe it, it appears to be more than an analogy to them.

Space and time are the desktop of our personal interface, and three-dimensional objects are icons on the desktop. Our interface gives the impression that it reveals true cause and effect… But this appearance of cause and effect is simply a useful fiction, just as it is for the icons on the computer desktop.

(The interface, to which they refer, is a ‘species-specific interface’, which means it’s a human consciousness interface. They don’t say if this interface applies to other sentient creatures, or just us.)

The issue of cause and effect being a ‘useful fiction’ was taken up by someone (authors of objections are not given) in objection 17, to which the authors of the theory responded thus:

Our views on causality are consistent with interpretations of quantum theory that abandon microphysical causality… The burden of proof is surely on one who would abandon microphysical causation but still cling to macrophysical causation.

I could respond to this challenge, but it’s not relevant to my argument. The point is that the authors obviously don’t ‘cling to macrophysical causation’, which I would contend creates a problem when discussing evolutionary theory. The point is that according to every discussion on biological evolution I’ve read, extant species are consequentially dependent on earlier species, which means there is a causal chain going back to the first eukaryota. If this causal chain is a ‘useful fiction’ then it is hard to see how any theory of evolution that excludes it could be called evolutionary. With or without this useful fiction, the authors ‘new theory’ turns evolution on its head, with conscious agents taking precedence over physical objects, including species, all of which are impermanent. In spite of this ontological difficulty, the authors believe that when they ‘project’ their ‘new theory’ onto the ‘species-specific interface’ of impermanent spacetime (which doesn’t exist unperceived), the old ‘standard theory of evolution’ will be found.

I’ve left a comment on the bottom of the web page (link given in intro above) which challenges this specific aspect of their theory (using different words). If I get a response I’ll update this post accordingly.

Dr Strange; a surprisingly philosophical movie

I have to admit I wouldn’t have gone to see this based on the trailer, as it just appeared to be a special effects spectacular, which is what you expect from superhero movies. And it seemed very formulaic - an apprentice, a mentor, a villain who wants to destroy the world - you know the script. What changed my mind was a review by Stephen Romei in the Australian Weekend Review (29-30 Oct. 2016), who gave it 3.5 stars, and re-reading it, gives a lot of the plot away. I’ll try not to do that here, but I’m not promising.

Dr Stephen Strange is played by Benedict Cumberbatch, who is much better cast here than in The Imitation Game, which I thought was a travesty. As an aside, The Imitation Game was an insult to the real Alan Turing, but I don’t believe that was Cumberbatch’s fault. I blame the director, writers and producers, who, knowing the audience’s ignorance, gave them the caricature of genius that they expected the audience wanted to see.

Cumberbatch’s Dr Strange is a self-obsessed, egotistical, unapologetically self-promoting brain surgeon. He’s never known failure and that’s an important psychological point in my view. The first subliminal philosophical reference in this movie is the well-worn trope: the unexamined life is not worth living. This is pretty much the theme or premise of every story ever told. The point is that no one examines their life until they experience failure, and, of course, Strange faces failure of a catastrophic kind. Otherwise, there’d be no movie.

He then goes on a mystical journey, which many of us may have done at an intellectual level, but can only be done viscerally in the world of fiction. I should point out that I went through a prolonged ‘Eastern philosophy’ phase, which more or less followed on from the ‘Christian’ phase of my childhood. I’m now going through a mathematical phase, as anyone reading this blog could not have failed to notice.

Anyway, Strange’s journey is distinctly Eastern, which is the antithesis of his medical-science background. But he is introduced to an ‘astral’ or ‘spirit’ dimension, and there is a reference to the multiverse, which is a current scientific trope, if I may re-use that term in a different context. I don’t mind that ‘comic book’ movies allude to religious ideas or even that they mix them with science, because one can do that in fiction. I’ve done it myself. The multiverse is an allusion to everything that we don’t know scientifically (even in science) and is the current bulwark against metaphysics. Employing it in a fantasy movie to enhance the fantasy element is just clever storytelling. It embodies the idea, that is still very current in the East, that science cannot tell us everything.

There are 2 mythological references in the movie, including one biblical one. At one point the villain, Kaecilius (played by Mads Mikkelsen) attempts to seduce Strange to the ‘dark side’, which is very reminiscent of Satan’s attempt to seduce Jesus in the desert. I’ve always liked that particular biblical story, because it represents the corruption of power and status over the need to serve a disenfranchised public. In other words, it is an appeal to ego over the need to subordinate one’s ego for a greater good.

One of the themes of the story is mortality and immortality; something I’ve explored in my own fiction, possibly more explicitly. We live in a time where, as Woody Allen once explained in literary terms, we ‘suspend disbelief’ that we are going to live forever. We tend to avoid, in Western culture, any reference to mortality, yet it is an intrinsic part of life. We all eventually get there but refuse to face it until forced to. This is actually addressed in this movie, quite unexpectedly, as we don’t expect lessons in philosophy in a superhero movie.

Last but not least, there is a subtle but clever allusion to Camus’ famous retelling of the Greek Sisyphus myth (look it up), not something your average cinema audience member would be expected to know. It is embedded in one of those plot devices that I love: where the hero uses an unexpected ‘twist’, both literally and figuratively, and where brain defeats overwhelming force.

Penrose's 3 Worlds Philosophy

This is the not-so-well-known 3 worlds philosophy of Roger Penrose, who is a physicist, cosmologist, mathematician and author. I’ve depicted them pretty well as Penrose himself would, though his graphics (in his books) are far superior to mine (and they don’t run off the page). I know it doesn’t quite fit, but if I made it fit it wouldn’t be readable.

Penrose is best known for his books, The Emperor’s New Mind and Road to Reality; the former being far more accessible than the latter. In fact, I’d recommend The Emperor’s New Mind to anyone who wants a readable book that introduces them to the esoteric world of physics without too many equations and lots of exposition about things like relativity, quantum mechanics, thermodynamics and cosmology. Road to Reality is for the really serious physics student and I have to admit that it defeated me.

The controversial or contentious part of Penrose’s diagram is the ‘Platonic World’ (Mathematics) and its relationship to the other two. The ‘Physical World’ (Universe) and the ‘Mental World’ (Consciousness) are not the least bit contentious - you would think - as everyone reading this is obviously conscious and we all believe that we inhabit a physical universe (unless you are a solipsist). Solipsism, by the way, sounds nonsensical but is absolutely true when you are in a dream.

I’ve mentioned this triumvirate before in previous posts (without the diagram), but what prompted me to re-visit it was when I realised that many people don’t appreciate the subtle yet significant difference between mathematical equations (like Pythagoras’s Theorem or Euler’s equation, for example) and physics equations (like Einstein’s E = mc² or Schrodinger’s equation). I’ll return to this specific point later, but first I should explain what the arrows signify in the graphic.

I deliberately placed the Physical World at the top of the diagram, because that is the intuitive starting point. The arrows signify that a very small part of the Universe has created the whole of consciousness (Penrose allows that it might not be all of consciousness, but I would contend that it is). Then a very small part of Consciousness has produced the whole of mathematics (that we know about) and here I would concede that we haven’t produced it all because there is still more to learn.

By analogy, according to the diagram, a small part of the Platonic (mathematical) world  ‘created’ the physical universe. Whilst this is implied, I don’t believe it’s true and I’m not sure Penrose believes it’s true either. Numbers and equations, of themselves, don’t create anything. However, the Universe, to all appearances and scientific investigations, is a consequence of ‘natural’ laws, which are all mathematical in principle if not actual fact. In other words, the Universe obeys mathematical rules or laws to an extraordinarily accurate degree that appear to underpin its entire evolution and even its birth. There is a good argument that these laws pre-exist the Universe (including critical constants of nature) and therefore that mathematics pre-existed the Universe, hence its place in the diagram.

So there are at least 2 ways of looking at the diagram: one where the Universe comes first and Mathematics comes last, or alternatively, Mathematics comes first and Consciousness comes last; the latter being more contentious.

I should point out that, for many philosophers and scientists, this entire symbolic representation is misleading. For them, there are not even 2 worlds, let alone 3. They would argue that consciousness should not be considered separately to the physical world; it is simply a manifestation of the physical world and eventually we will create it artificially. I am not so sure on that last point, but, certainly, most scientists seem to be of the view that artificial intelligence (AI) is inevitable and if it’s indistinguishable from human intelligence then it will be conscious. In fact, I’ve read arguments (in New Scientist) that because we can’t tell if someone else has consciousness like we do (notice that I sabotaged the argument by using ‘we’) then we won’t know if AI has consciousness and therefore we will have to assume it does.

But aside from that whole other argument, consciousness plays a very significant role, independently of the Universe itself, in providing reality. Now bear with me, because I contend that consciousness provides an answer to that oft asked fundamentally existential question: why is there something rather than nothing? Without consciousness there might as well be nothing. Think about it: before you were born there was nothing and after you die there will be nothing. Without consciousness, there is no reality (at least, for you).

Also, without consciousness, the concepts of past, present and future have no relevance. In fact, it’s possible that consciousness is the only thing in the Universe that exists in a continuous present, which means that without memory (short term or long term) we wouldn’t even know we were conscious. I’ve made this point in another post (What is now?) where I discuss the possibility that quantum mechanics is in the future and so-called Classical physics is always in the past. I elaborate on a quote by Nobel laureate, William Lawrence Bragg, who effectively says just that.

Not to get too far off the track, I think consciousness deserves its ‘special place’ in the scheme of things, even though I concede that many would disagree.

So what about mathematics: does it also deserve a special place in the scheme of things? Most would say no, but again, I would say yes. Let me return briefly to the point I alluded to earlier: that mathematical equations have a different status to physics equations. Physics equations, like E = mc², only have meaning in reference to the physical world, whereas a mathematical equation, like Euler’s equation, e^ix = cos x + i sin x, or his more famous identity, e^iπ + 1 = 0, have a meaning that’s independent of the Universe. In other words, Euler’s identity is an expression of a mathematical relationship that would still be true even if the Universe didn't exist.

Again, not everyone agrees, including Stephen Wolfram, who created Mathematica, so certainly much more clever than me. Wolfram argues, in an interview (see below) that mathematics is a cultural artefact, and I’ve come across that argument before. Wolfram has also suggested, if my memory serves me correctly, that the Universe could be all algorithms, which would make mathematics unnecessary, but I can’t see how you could have one without the other. Gregory Chaitin, quotes Wolfram (in Thinking about Godel and Turing) that the Universe could be pseudo-random, meaning that it only appears random, which would be consistent with the view that the Universe is all algorithms. Personally, I think he’s wrong on both counts: the Universe doesn’t run on algorithms and it is genuinely random, which I’ve argued elsewhere.

The problem I have with mathematics being a cultural artefact is that the more you investigate it the more it takes on a life of its own, metaphorically speaking. Besides, we know from Godel’s Incompleteness Theorem that mathematics will always contain truths that we cannot prove, no matter how much we have proved already, which implies that mathematics is a never-ending endeavour. And that implies that there must exist mathematical ‘truths’ that we are yet to discover and some that we will never know.

Godel’s Theorem seems to apply in practice as well as theory, when one considers that famous conjectures (like Fermat’s Last Theorem and Riemann’s Hypothesis) take centuries to solve because the required mathematics wasn’t known at the time they were proposed. For example, Riemann first presented his conjecture in 1859 (the same year Darwin published The Origin of Species), yet it has found connections with Hermitian matrices, used in quantum mechanics. Riemann’s Hypothesis is the most famous unsolved mathematical problem at the time of writing.

The connection between mathematics and humanity is that it is an epistemological bridge between our intellect and the physical world at all scales. The connection between mathematics and the Universe is more direct. There are dimensionless numbers, like the fine-structure constant, the mass ratio between protons and neutrons and the ratio of matter to anti-matter, all of which affect the Universe's fundamental capacity to produce sentient life. I wrote about this not so long ago. There is the inverse square law, which is a mathematical consequence of the Universe existing in 3 spatial dimensions that allows for extraordinarily stable orbits over astronomical time frames. Then there is quantum mechanics, which appears to underpin all of physical reality and can only be revealed in the language of mathematics.

Footnote 1: Stephen Wolfram's argument that mathematics is a cultural artefact and that there is no Platonic realm. Curiously, he uses the same examples I do to come up with a counter-argument to mine. I mostly agree with what he says; we just start and arrive at different philosophical positions.

Footnote 2: This is Roger Penrose being interviewed by the same person on the same topic, and giving the antithetical argument to Wolfram's. You can see that he and I are pretty well in agreement on this subject.

Footnote 3: This is Penrose's own take on his 3 worlds.

The relationship between science and philosophy

I’ve written on this before, but recent reading has made me revisit it, because I think it’s a lot closer and interrelated than people think, especially among scientists. I’m referring to the fact that more than one ‘famous’ scientist has been dismissive of philosophy and its contribution to our knowledge. I’m thinking Richard Dawkins, Stephen Hawking, Peter Atkins and, of course, Richard Feynman, whom I particularly admire.

In the Western epistemic canon, if I can use that term, philosophy and science have a common origin, as we all know, with the Ancient Greeks. There was a time when they were inseparable, and certainly up to Newton’s time, science was considered, if not actually called, ‘natural philosophy’. In some circles, it still is. This is to distinguish it from metaphysics, and I think that division is still relevant, though some may argue that metaphysics has no relevance in the modern world.

Plato argued that ‘Metaphysics… holds that what exists lies beyond experience’ (my on-board computer dictionary definition) which in the Platonic tradition would include mathematics, oddly enough. But in the Kantian and Hume tradition: ‘…objects of experience constitute the only reality’ (from the same source).  I would suggest that this difference still exists in practice if not in theory. In other words, science is based on empirical evidence, though mathematics increasingly plays a role. Mathematics, by the way, does not constitute empirical evidence, but mathematics constitutes a source of ‘truth’ that can’t be ignored in any assessment of a scientific theory.

I find I’m already heading down a path I didn’t intend to follow, but maybe I can join it to the one I intended to follow further down the track. So let me backtrack and start again.

Most scientific theories start off in the realm of philosophy, though they may be informed by limited physical evidence. Think, for example, of Darwin’s theory of evolution by natural selection. Both he and Alfred Wallace (independently) came to the same conclusion, when they traveled to little-known parts of the world and saw creatures that were not only exotic but strange and unexpected. Most significantly, they realised how geography and relative isolation drove species’ diversity. This led them both to develop an unpopular and unproven philosophy called evolution. Evidence came much later in the form of fossils, genetics and, eventually, DNA, which is the clincher. Evidence can turn philosophy into science and theories into facts.

As anyone, who has any exposure to American culture, knows, the philosophical side of this debate still rages. And, to some extent, this is the very reason that some scientists would argue that philosophy is irrelevant or, at the very least, subordinate to science. This point alone is worth elaborating on. There is a dialectic between science and philosophy and the dominant discipline, for want of a better term, is simply dependent on our level of knowledge, or, more importantly perhaps, our level of ignorance. By dialectic I mean a to-ing and fro-ing, so that one informs the other in a continual and constructive dialogue, which leads to an evolvement which we call a theory.

Going back to the example of the theory of evolution, which, after 150 years, is both more fraught with difficulties and more cemented in evidence than either Darwin or Wallace could have imagined. In other words, and this is true in every branch of science, the more we learn about something the more mysteries we uncover. For example, DNA reveals in extraordinary relief how every species is related and how all life on Earth had a common origin, yet the origin and evolution of DNA itself, whilst not doubted, poses mysteries of its own. And while mysteries will always exist, anti-science proponents will find a foothold to sow scepticism and disbelief.

But my point is that the philosophy of evolutionary biology is strengthened by science to the extent that it is considered a fact by everyone except those who argue that the Bible has more credibility than science. Again, I’m getting off-track, but it illustrates why scientists have a tendency to demote philosophy, when it is used to promote ignorance over what is already known and accepted in mainstream science.

On a completely different tack, it’s well known that Einstein held a deep scepticism about the validity and long-term scientific legacy of quantum mechanics. What is lesser known is his philosophical belief in determinism that led him to be so intractable in his dissent. Einstein’s special theory of relativity led to some counter-intuitive ideas about time. Specifically, that simultaneity is subjective, not objective, if events are spatially separated (refer my post on Now). Einstein came to the philosophical conclusion that the Universe is determinant, where space and time are no longer separate but intrinsically combined in space-time. Mathematically, this is resolved by treating time as a fourth dimension, and, in Einstein’s universe, the future is just as fixed as the past, in the same way that a spatial dimension is fixed. This is a philosophical viewpoint that arose from his special theory of relativity and thus informed his worldview to the point that it contradicted the inherent philosophy of quantum mechanics that tells us, at a fundamental level, everything is random.

And this brings me full circle, because it was reading about the current, increasingly popular, many-worlds interpretation of quantum mechanics that led me to contemplate the metaphorically and unavoidably incestuous relationship between philosophy and science. In particular, adherents to this ‘theory’ have to contend with their belief that every action they do in this universe affects their counterparts in parallel universes. I’ve expressed my dissent for the many-worlds interpretation of quantum mechanics elsewhere, so I won’t discuss it here. However, I would like to address this specific consequence of this specific philosophy. You have a stream of consciousness that is really the only thing you have that gives you a reality. So, even if there are an infinite and continual branching of your current universe into parallel universes, your stream of consciousness only follows one and axiomatically that’s the only reality you know.

And now, to rejoin the path that led me astray, let's talk about mathematics. Mathematics has followed its own historical path in Western thought alongside science and philosophy with its own origins in Plato’s Academy. In fact, Plato adopted the curriculum or quadrivium from Pythagoras’s best student, Archytas (after specifically seeking him out), which was arithmetic, geometry, astronomy and music. Mathematics is obviously the common denominator in all these.

Mathematics also has philosophical ‘schools’ which I’ve written about elsewhere, so I won’t dwell on that here. Personally, I think mathematics contains truths that transcend humanity and the universe itself, but it’s the pervasive and seemingly ineluctable intrusion into science that has given it its special epistemological status. String Theory or M Theory is the latest, most popular contender for a so-called Theory of Everything (TOE) yet it’s more philosophy than scientific theory. It’s only mathematics that gives it epistemic status, and it’s arguably the best example of the dialect I was talking about. I’ve written in another post (based on Noson Yanofsky’s excellent book) that we will never know everything there is to know in both science and mathematics. This means that our endeavours in attempting to understand the Universe (or multiverse) will be never-ending, and thus the dialectic between science and philosophy will also be never-ending.