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 (as Paul Davies has proposed) 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, eix = cos x + i sin x, or his more famous identity, eiπ + 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.