I’d be shocked if you’ve ever come to this site and not once wondered what’s going on with that picture across the top of each page. Who is this old scholarly dude and why is he looking at me? What are the balls all about? Well, this post is meant to give you my own reasoning for it, and interpretation of it. You’re certainly free to have your own interpretation, and if yours differs from mine I’d love to hear about it! Accordingly, this post will be purely philosophical, but that’s why it’s a post and not a lesson—I reserve the blog posts for shamelessly waxing philosophical.
The picture is a cut-off version (because WordPress.com can’t fit the whole thing) of M.C. Escher’s “Three Spheres II”. I won’t explain the picture, because there’s nothing I could say about it that you couldn’t deduce by looking at it (“there are three spheres, one is white,…”). First and foremost, if you’re not familiar with Escher’s work, Google image him right now because the guy was a total genius. Many of his prints are highly motivated by mathematical ideas, and some of these ideas are rather sophisticated (hyperbolic geometry, periodic tilings, impossible shapes, etc.).
This print in particular is, or at least I view it to be, a representation of Roger Penrose’s extremely gorgeous philosophy of mind, math, and physics. In his view of things, the world is divided up into three “smaller” worlds: the mathematical world, the physical world, and the conscious world.
The mathematical world can be viewed in a similar way as one views Plato’s world of forms—i.e., of supreme and eternal perfection, containing the truest essence of things. The only difference now is that this world of “forms” is a purely mathematical one; our forms are now nothing but mathematical structures.
The physical world is precisely what it sounds like: the world consisting of physical objects like your computer, this cup of coffee I’m drinking, stars, galaxies, black holes, atoms, and your body. This is the world that knows about the laws of physics, and has all of its constituents abide by them.
Lastly, the conscious world is that which consists of those things which are pure manifestations of consciousness. Seeing as consciousness is still not very well understood and extremely difficult to define, I’ll simply leave the definition to whatever the reader feels is that “obvious” sense of “being conscious” that we all (presumably) experience.
The key idea behind this philosophy is that these three worlds are not independent of each other. Instead, they influence and interact with each other in ways that are hard to deem as anything other than miraculous. In particular, the conscious world is able to interact with the mathematical world, since (assuming that humans are conscious) we use our consciousness to discover various mathematical structures and to understand their various relationships with each other. I.e., when we “do math”, we’re really using our consciousness to access this “other world” of eternal mathematical structures. Additionally, the mathematical world influences the physical world due to the fact that, as far as we can see, there is mathematical consistency to the laws that govern the physical world. I.e., the physical world is somehow nothing but a “manifestation” of certain structures that exist in the mathematical world. Lastly, the physical world affects the conscious world because (again, only as far we currently know), conscious things require a physical background to be implemented. I.e., whatever it is that makes us (or anything) conscious lies somewhere in our brains, and/or our bodies, and/or somewhere in the physical universe. In this way, these three worlds are completely interconnected in precisely the way that figure 1 depicts.
Figure 1 (Drawn by Roger Penrose himself)
The reason I find this philosophy so pleasing is that each link interconnecting one world to the next is in no way obvious, or necessary. I cannot think of, nor have I encountered, any convincing argument for why any of these links had to be there. There is no reason that the laws of physics have to be mathematically consistent (assuming they are), or even that mathematics would be the proper language for describing them. One could plausibly imagine a world of genuine chaos (and indeed, some people do believe this to be our own physical world at the deepest levels, for which there is no evidence to either support or deny at this point). Moreover, it is in no way obvious that consciousness, whatever it is, has to use some physical object (like a brain) as a medium. We could perfectly well imagine some consciousness “floating around” out there which is not “tied down” to a physical embodiment. And lastly, there’s no a priori reason why anything that is conscious has to be able to know how to do math. I.e., it is perfectly reasonable to assume that something could be conscious without ever being able to develop some sophisticated logical machinery through which to access the world of math (and therefore the world of physics, by the transitive property of figure 1). The fact that all of these links appear to exist, at least to an extent, is incredible (in my view).
Note, however, that there are more to these worlds than their ability to influence “the next” world, in that there is more that our consciousness can do than just math, and there is more that math can do than just describe the laws of physics, and there is more that the physical world can do than just give rise to consciousness. This is embodied in figure 1 by the fact that only a part of each sphere is used to “hit” the next sphere.
Now let’s get back to the picture on each page of the site. What I see (and feel free to disagree) are these three worlds, sitting on M.C. Escher’s desk. The middle sphere, containing a reflection of Escher, is the conscious sphere (this is obvious, since it’s the one with a human “in” it!). The sphere on the left (as viewed from the artist) is the physical world, as it appears to be made of glass, a very “physical” object. And the sphere on the right is the mathematical world—completely clean and perfect. Moreover, we can see the reflection of both of these worlds in the middle sphere, which is the conscious world. In my mind, this signifies the fact that while the arrow goes from consciousness to mathematics, we can still posit the existence of the physical world via our conscious world. In other words, each world “knows about” the other two, but we can only “see” out of the conscious world, as that is in some clear way the world that we interact with the most.
I could write forever on this, but I think I’ll stop here. Now you at least have some explanation as to why this picture was chosen—whether you agree with this interpretation or not is entirely up to you. I have no pretentions of believing that this is anything but an untestable and unprovable philosophy. Regardless, this set of ideas has given me a very meaningful relationship with my craft (mathematical physics), and maybe others will find some beauty in this way of viewing things as well!
The True Beauty of Math 
love when you wax philosophic, and the artwork of escher is cool! keep it coming it’s BEAUTY-ful!
Does the physical world affect our conscious world as well as our mathematical world? Most obviously, our consciousness is due to the fact that we have brains, hearts, and other organs to keep us alive. As for the mathematical reality, isn’t something like number and quantity due to the fact that we can visualize things as distinct things? If our perception of the world were just one big blur its hard to imagine that we would conceive anything like numbers.
Well it’s all about the direction that the arrows point in (in Penrose’s drawing). Namely, it is indeed the physical world that allows the mental world to be accessed, as you mentioned. Similarly, it’s the mental world that allows the mathematical world to be accessed. Note that this doesn’t mean that the mental world CREATES the mathematical world. I.e., the mathematical world would (in this philosophy) still exist regardless of whether or not there were a mental world, and regardless of what form the mental world took (i.e., if our view of the world still existed). The philosophy suggests that all three worlds are already “out there” in an objective sense, and that remarkably they are connected (at least partially) in the way described. No world “creates” any part of the other, they all just let us “see” (in the appropriate sense) the others.
The Meaning of Life
===============
In the primordial soup of early Earth, a molecule happened to interact with it’s neighbours in such a way as to result in other molecule just like it being formed. Now as you can imagine, there are many more reactions that don’t do this than those that do; but with lots of interaction over lots of time, it was bound to happen. Now simply by the fact that this molecule resulted in more like it being created (‘formed’ would perhaps be the better word here as matter is not springing into existence), it naturally increased in ‘popularity’ in the soup. Through random changes, other ‘replicators’ emerged too and eventually formed us (there was a lot of luck involved, so we’re not really the “pinnacle of creation/evolution” or “bound for supremacy” as much as we’d like to think so).
Richard Dawkins put it more eloquently than me in his book “The Selfish Gene”, which I’d recommend. Having read “A Drunkard’s Walk” just a few days ago, it’s apparent that humans sometimes try to find meaning where there is not, and I have no qualms about this interpretation of the meaning of life: reproducing systems. The question of the purpose of life then becomes meaningless: from the initial configuration of the universe, atoms collided in ways (some of which we can describe using mathematical formulas, see later) that resulted in us. There has even been progress in recreating some of those initial conditions in a bottle, and seeing such processes occur.
Reality check: do I have qualms with the concept of not having a soul and ‘me’ really just being a special configuration of particles with no ulterior purpose? Sometimes I wake up in a sweat, but on the whole: not really, that’s just the way it is.
The Meaning of Consciousness
========================
Onto consciousness then. This is proving to be much more difficult to quantify; but as neuroscience is progressing, we’re finding out more and more about the ‘brain’ leaving less and less space and functions for the ‘mind’ (to borrow terms from this book I happened to read randomly about loonie bins called “Girl, Interrupted”).
Still, what is consciousness? Pursuing dictionary definitions leads to a circular loop between ‘awareness’ and ‘perception’. Perhaps perception could just be understood mechanically (or electrically or physically or whichever word you prefer) as an interaction from the external world simply resulting in an internal change of state? This definition applies to rocks to (any force must have changed some property of the rock particles), which is problematic as we wouldn’t consider rocks to perceive; but perhaps the only difference with life is that the internal systems in place respond much more complexly.
So what does it mean to think? *consults a dictionary again* No help there; but I’d say just internal communication within that complex multi-state system. I’ll leave this to ‘real’ philosophers to wax about what is means or requires to wonder about one’s self.
† If you’re willing to suspend belief long enough for a novel, read the series ‘His Dark Materials’ (of which ‘ The Golden Compass’ is a part); the novel may seem geared toward’s children from the cover, but its take on consciousness and authority was fascinating.
The Meaning of Mathematics
======================
You know the routine by now. After initially having written a para or two, similarities to one of Feynman’s lectures titles “The Meaning of it All” (which is a very concrete of lectures, and not just masturbatory speculating) became apparent; so I’ll just start quoting him (taken form the [first part](http://www.inf.fu-berlin.de/lehre/pmo/eng/Feynman-Uncertainty.pdf)).
> What is science? The word is usually used to mean one of three things, or a mixture of them. I do not think we need to be precise—it is not always a good idea to be too precise. Science means, sometimes, a special method of finding things out. Sometimes it means the body of knowledge arising from the things found out. It may also mean the new things you can do when you have found something out, or the actual doing of new things. This last field is usually called technology—but if you look at the science section in Time magazine you will find it covers about 50 percent what new things are found out and about 50 percent what new things can be and are being done. And so the popular definition of science is partly technology, too.
>
> I want to discuss these three aspects of science in reverse order. […]
So do I. The equivalent of technology here would be ‘applied math’. I think math lovers will instinctively wince at the sight of that term due to memories of teachers having made them think that that was was math was all about almost having turned them away from it. That’s not to say it is not crucial or that it’s not respectable:
> I will begin with the new things that you can do—that is, with technology. The most obvious characteristic of science is its application, the fact that as a consequence of science one has a power to do things. And the effect this power has had need hardly be mentioned. The whole industrial revolution would almost have been impossible without the development of science. The possibilities today of producing quantities of food adequate for such a large population, of controlling sickness—the very fact that there can be free men without the necessity of slavery for full production—are very likely the result of the development of scientific means of production.
Moving on:
> The next aspect of science is its contents, the things that have been found out. This is the yield. This is the gold. This is the excitement, the pay you get for all the disciplined thinking and hard work. The work is not done for the sake of an application. It is done for the excitement of what is found out. Perhaps most of you know this. But to those of you who do not know it, it is almost impossible for me to convey in a lecture this important aspect, this exciting part, the real reason for science. And without understanding this you miss the whole point. You cannot understand science and its relation to anything else unless you understand and appreciate the great adventure of our time. You do not live in your time unless you understand that this is a tremendous adventure and a wild and exciting thing.
>
> […] And the newspapers, as you know, have a standard line for every discovery made in physiology today: “The discoverer said that the discovery may have uses in the cure of cancer.” But they cannot explain the value of the thing itself.
The last bit, what ‘doing mathematics’ really means, diverges a bit;
> The third aspect of my subject is that of science as a method of finding things out. This method is based on the principle that observation is the judge of whether something is so or not. All other aspects and characteristics of science can be understood directly when we understand that observation is the ultimate and final judge of the truth of an idea.
>
> […] The principle that observation is the judge imposes a severe limitation to the kind of questions that can be answered. They are limited to questions that you can put this way: “if I do this, what will happen?” There are ways to try it and see. Questions like, “should I do this?” and “what is the value of this?” are not of the same kind.
>
> […] Another very important technical point is that the more specific a rule is, the more interesting it is. The more definite the statement, the more interesting it is to test. […]
Clearly, observation of the physical world is meaningless to mathematical ideas. Here, I believe, consistency is the ultimate test. Now an interesting question here is: does the exception prove the rule here too? Or to be more precise, can we prove something is consistent rather than just be very convinced that it is until someone is able to come up with a contradiction like what happens in science? Is all of math damned to be provisional too, like naive set theory was?
Googling turned up this Math.StackExchange question titled [How to prove an axiom system is consistent?](http://math.stackexchange.com/q/120016/30375); but I’m not smart enough to grok everything. This is what I think: unless your system of axioms includes that it is consistent as an axiom itself, we can only say that it is consistent if ZFC is consistent. But Gödel’s second incompleteness theorem says that ZFC itself cannot be used to prove that it is consistent (i.e., we only have relative and not absolute consistency)!
There’s also some similarity between Gödel’s first incompleteness theorem and the uncertainty principle of Quantum Mechanics: we cannot know *everything* about a (mathematical or physical) system for sure.
The 3 ‘worlds’
==========
This cyclically dependent worlds seems much like the Hindu concept of Karma or cosmic revenge: it’s convenient/beautiful to believe, but I don’t subscribe to this idea of math forming an ‘intangible eternal blah blah world’. TBH, I feel it’s much like arguing about the existence of God (I was afraid to bring up religion, but now that I’ve let the beast out…): since we really have very vague and often incompatible definitions about God, there is no meaning to the debate until it is pinned down; that is the reason I position myself as a theological non-cognitivist (sorry for the incoming spam on your blog, BTW), and feel talking about these ‘realms’ is pointless too. Our imagination is just thoughts occurring without accessing any special ‘world’ which theorems have to inhabit. Where do ideas come from?
> In that sense it makes no difference where the ideas come from. Their real origin is unknown; we call it the imagination of the human brain, the creative imagination—it is known; it is just one of those “oomphs.”
Or the result of those complex systems we were talking about. I’m still amazed my trillions of inter-connected neurons can do this:
> It is surprising that people do not believe that there is imagination in science. It is a very interesting kind of imagination, unlike that of the artist. The great difficulty is in trying to imagine something that you have never seen, that is consistent in every detail with what has already been seen, and that is different from what has been thought of; furthermore, it must be definite and not a vague proposition. That is indeed difficult.
As for the influence of mathematics on the physical world — actually you know what: I’ll end this sentence right here and try again. *takes a breath* The universe behaves as it does; and again, I’m going to let the professional philosophers waste their time thinking about where it came from and all. All we know is that it exists, and to a certain extent how it exists thanks to our observations. We just noticed that we could describe and predict a lot of stuff that goes on using these abstract formulas.
Could it perhaps be fundamentally non-mathematical at its most fundamental level? Let’s try to pin what we really mean by ‘mathematical’ in the previous sentence: the first and third definitions clearly don’t apply, so we’re left with “the body of knowledge we’ve accumulated so far in using the mathematical method”. So if it doesn’t, all that means is that we haven’t developed the math to describe it yet (which is less absurd when you realize that the average grad student today, equipped with the ground-breaking concepts of negative numbers and calculus, has more as his disposal than Newton’s ancestors did).
My rant is beginning to feel unnecessary (“What? *Now* you realize that?!”), so I’ll end on a more romantic note:
> First, there was the earth without anything alive on it. For billions of years this ball was spinning with its sunsets and its waves and the sea and the noises, and there was no thing alive to appreciate it. Can you conceive, can you appreciate or fit into your ideas what can be the meaning of a world without a living thing on it? We are so used to looking at the world from the point of view of living things that we cannot understand what it means not to be alive, and yet most of the time the world had nothing alive on it. And in most places in the universe today there probably is nothing alive.