IMHO a must-read (especially for science writers and journalists):
On Math, Matter and Mind
Foundations of Physics 36 (2006), pp. 765-794
physics/0510188
A Mystic (Piet Hut), a Secularist (Mark Alford), and a Fundamentalist (Max Tegmark) discuss the nature of reality in the ontological context of Roger Penrose’s math-matter-mind triangle, according to which Matter somehow embodies Math, the Mind arises from Matter, and Math is a product of the Mind.
From the paper’s conclusion:
A key message for non-physicists reading this paper is… that they should be deeply suspicious of any self-proclaimed popularizer or other ambassador claiming to speak on these matters on behalf of the consensus of the theoretical physics community.
Summary:
According to the Fundamentalist, Math is the foundation, Mind is the “feel” of information being processed, and there should be no Mind-Math link since Math is not a creation of the human mind.
According to the Secularist, there should be only the Mind-Math link since Math is a creation of the human mind.
According to the Mystic, the three M’s co-emerge from a source beyond the three M’s, and the three arrows of Penrose’s diagram should be replaced by a different trio:
The Fundamentalist speaking for himself:
After Wigner had written his famous essay on the unreasonable effectiveness of mathematics, the standard model of particle physics revealed new “unreasonable” mathematical order in the microcosm of elementary particles, and my guess is that history will repeat itself again and again. I know of no other compelling explanation for this trend than that the physical world really is completely mathematical, isomorphic to some mathematical structure.
Let me briefly elaborate on what I mean by this hypothesis that mathematical and physical existence are equivalent. It can be viewed as a form of radical Platonism, asserting that the mathematical structures in Plato’s realm of ideas… exist in a physical sense… However, it is crucial to distinguish between two ways of viewing a physical theory: from the outside view of a physicist studying its mathematical equations, like a bird surveying a landscape from high above it, and from the inside view of an observer living in the world described by the equations, like a frog living in the landscape surveyed by the bird. From the bird perspective, the physical world is a mathematical structure, an abstract, immutable entity existing outside of space and time. If history were a movie, the structure would correspond not to a single frame of it but to the entire videotape.
Consider, for example, a world made up of pointlike particles moving around in three-dimensional space. In four-dimensional spacetime — the bird perspective — these particle trajectories resemble a tangle of spaghetti. If the frog sees a particle moving with constant velocity, the bird sees a straight strand of uncooked spaghetti. If the frog sees a pair of orbiting particles, the bird sees two spaghetti strands intertwined like a double helix. To the frog, the world is described by Newton’s laws of motion and gravitation. To the bird, it is described by the geometry of the pasta — a mathematical structure. The frog itself is merely a thick bundle of pasta, whose highly complex intertwining corresponds to a cluster of particles that store and process information…
It took the genius of Einstein to realize that frogs living in Minkowski space would perceive time to slow down at high speeds, and that of Everett to realize that a single deterministically evolving quantum wavefunction in Hilbert space contains within it a vast number of frog perspectives where certain events appear to occur randomly.
I believe that consciousness is the way information feels when being processed… The human mind… emerges from math, as a self-aware substructure of an extremely complicated mathematical structure. Each such substructure subjectively perceives itself as existing in a physically real sense. Given the mathematical equations that describe our Universe, an infinitely intelligent mathematician could in principle deduce the properties of both its material content and the minds of its inhabitants.
The Secularist speaking for himself:
Everyone should take science seriously, while remembering that it is a human creation, not an all-embracing metaphysics.
There is a wide range of entities that are posited in physical theories, from fairly concrete ones like electric fields to more abstract ones like an object’s wave function, or its center of gravity, that we don’t (all) think of as “existing” or “being real” in the normal sense of those words. So just because something plays a role in our theorizing doesn’t mean it is a real object. And when a physicist argues that something is real he or she invokes a specific instance where the object is present: “of course electric fields are real: you can tell that there’s an electric field around a proton because it pushes other positive charges away.” One couldn’t speak of Hilbert space or the complex plane as giving away its presence in a similar way.
Although mind and matter are separate in this diagram, I do not want to imply that they are disjoint (dualism). I see them as two aspects of the world that we happen to distinguish quite sharply…
As it becomes harder to obtain experimental data against which to test our most fundamental theories, it will become harder to know whether we should believe them. One cannot predict how fundamental physics will then progress. It might continually find unexpected directions in which to proceed, such as the one hoped for by the Mystic. It might change so drastically that we would no longer recognize it as physics, or even science. Finally, it might simply atrophy away.
If I grapple with any deep mysteries, they are not about the meaning of science, but about the way people see the world. Science is the process of building a picture of the world that is, where possible, quantitatively accurate… But many people feel that science is incomplete without a deeper claim than mere predictive power. They want to tilt the balance towards conservatism, treating the picture, current or future, as metaphysical truth… It is striking, and mysterious to me, that so many people feel a strong need for such a frozen picture. What impresses me is the degree to which our current understanding of the universe would be utterly incomprehensible to previous generations. It seems very reasonable to expect that human ideas will continue to develop in unpredictable ways, and that the theories of the future will explore directions that we cannot even imagine today.
The Mystic speaking for himself:
I expect that science will ultimately give us profound insights into the real nature of the world. But unlike the Fundamentalist, I believe these will not emerge in any straightforward way from science as it is currently constituted. Rather, I expect science to metamorphose into something so different that it is literally inconceivable for us. So in that sense I agree with the Secularist that physics will probably see upheavals even (far) more fundamental than the discovery of quantum mechanics. And I agree with the Fundamentalist that Science will ultimately come arbitrarily close to a full understanding of reality.
The reason I like the word ‘mystic’ is that the future science I envision will be so different from current science, and the role of elements such as math and experiments will be so different from what they are now, that we have not the foggiest idea of what these will look like. The structures of a future science will remain a mystery, and the only thing we can be pretty sure of is that our current lines of reason will be seen to be naive and superficial, compared with the newer and deeper insights.
As for the Penrose diagram, I have deep doubts about all the links. Making these links now, before a future unification, seems premature. I strongly believe that the process of unification, which has successfully uncovered intrinsic links between, e.g., electricity and magnetism, space and time, matter and energy, will continue… And one example may well involve our three M aspects, matter, mind, and math. These three can then no longer can be treated as independent notions that have the power to point to each other. Drawing arrows, in my view, is simply a precursor to the program of unification, in which nature is discovered to be already unified more than we had thought…
I do not want to buy into an unquestioned prior status of the object pole over the subject pole of experience… I do not deny that a deep understanding of the material structure of the human brain will shed a lot of light on the way in which we experience; but the very fact that we experience may completely fall outside such an explanatory framework. My guess would be that such a question requires a shift to a wider horizon of knowledge/meaning/explanation… It is too simple to say: no brains no experience, hence brains produce experience…
The proper defense of all three links is not by ascribing to them any power in terms of causality, but by pointing to meaningful correlations that exist between the three M-elements. To take the example of a movie: within the movie, all kind of phenomena seem to ’cause’ other phenomena, in rather precise ways, but we know that the real cause lies in the projector, and in the process of shooting the film in the first place…
Among the three of us, I am the only one who can accept all three links, without being bothered by the circular nature of the ‘vicious triangle’ links. Since I only ascribe relative meaning to them, I have no need for any foundation. None of the elements within the story of reality is absolute or basic; all elements emerge simultaneously from a deeper unification, speaking in physics lingo. In Buddhist lingo, for example, this could be called co-dependent arising…
The resistance that mathematical objects show to our attempts to prove what later turns out to be false is akin to the resistance that physical objects show when we try to wish them away — both appear to have an existence independent of the presence or absence of individual humans. In this sense, I am sympathetic to [the] notion of archaic mathematical reality as being as real as physical reality…
In practical terms, it makes sense to deal with the world around us in terms of material objects and energetic processes, and it also makes sense to treat our experience as something that has equal pride of place. The problem arises when we try to isolate elements from this story, and point to some of them as truly fundamental. In my view, a future physics will transcend any story we have woven so far…
In my alternative picture, “?” stands for an origin that cannot be easily described, the way each of the other three can. Our three M’s are more like the shadows on the wall of Plato’s cave… The Source or Origin lies beyond that, and is more real than any particular element of what we conventionally take reality to be… Note here that in fact, upon finer scrutiny, the separation between “?” and the three M’s is only illusory. The real mind cannot be captured in a description, nor can the real matter. I’d say that even the real math cannot, if you include the living intuitive process of discovery…
We have painted ourselves in a corner, scientifically, by describing the whole world in objective terms, and finding less and less room for ourselves to stand on. The challenge we now face is not to reduce ourselves also to objects, but to explore ways to let science naturally widen its area of investigation, while staying true to its methodology of peer review, based on an interplay between theory and experiment, with experiment having the last word.
In various areas of science, from quantum mechanics to neuroscience and robotics, the subject pole of experience can no longer be neglected. Most likely, science will change qualitatively with this required extension of its methodology. This will not happen overnight. I expect this program to be carried out over a period of time comparable to the time it took to get the science of the object up and running, perhaps a century, quite likely a few centuries. But this process cannot and should not be hurried by wishful thinking or by external agendas… Real progress in physics can only come from within, from a necessity to introduce wider frameworks of explanation and interpretation to accommodate experimental facts that cannot be satisfactorily dealt with in the existing frameworks.
Mutual critiques:
The Secularist’s critique of the Mystic:
The Mystic thinks we will come to [know the ultimate constituents of the world] in the future… When is the right time to promote whatever picture we have found workable to metaphysical status?
I do not think it is meaningful to invoke a hidden or ineffable source. If it cannot even be described, what role can it play in our quest for knowledge?
I do not think that math deserves to be put on the same footing as mind and matter… What determines which human activities we include as separate poles connected directly to the ineffable origin? Why not include language, music, art, and so on?
The Mystic’s response:
I see science as converging further and further toward what is true about reality, but I don’t expect there to be a particular point at which we can declare victory. The road toward deeper insight may be unending, simply because the degrees of what can be called insight may be inexhaustible… My main point is not that we will reach a final truth, but rather that we will never reach an absolute boundary beyond which we cannot progress in getting closer to the truth.
As for the ineffability of a source, that is just a relative notion. When all you have are concepts such as electricity and magnetism, then electromagnetism is ineffable, since it doesn’t fit into your framework. It has to be invoked as a new type of source that can be projected in your existing framework, very much like the shadows in Plato’s cave. But then you enlarge your framework in order to include the new source.
The requirement that any source should be describable is not something I want to buy into… I expect a future science to throw out some of what we now still see as absolutely essential for the scientific method…
Under matter we take anything physical, such forms of pure energy like radiation, or even the vacuum nowadays. Similarly, under math we should take any conceptual structure that we use in our mind in order to make sense of both mind and (material) world. Language, logic of all sorts (including common sense reasoning), any distinctions that are found to be useful I would group here under “math”…
“Math” stands out as the surprising fact of the regularity of the world, or if you like, the surprising fact that we can view the world in such unexpectedly regular ways and get away with it, without contradictions, and with the ability to make accurate predictions.
The Fundamentalist’s critique of the Mystic:
My main objection to both the other source view and the no-source view is an admittedly emotional one: I perceive them as defeatist… Both indicate that the quest for a fundamental theory of everything is ultimately doomed, and that the best we can do is go off and meditate or optimize a ball bearing. Although the Mystic hopes for great conceptual breakthroughs ahead, I am not persuaded by his hypothesis that we will come arbitrary close to a complete understanding while remaining precluded from ever attaining it. I also find the Mystic battle plan disturbingly vague, failing to understand what specific steps we should take to explore the object pole.
The Mystic’s Response:
The Fundamentalist is in effect looking for the lost key under the lamp post. He takes the objectivistic program, in which everything is explained in third-person terms as interacting objects, and hopes that that program will carry the day. But I see that as a limiting case of a wider program, in which first-person experience and a study of the subject qua subject will augment our already very detailed studies of objects. Within that wider program, I think we have much more of a chance to find a horizon in which we can see how all of our experiences hang together in a meaningful way, without artificially reducing everything to a particular subset of privileged phenomena.
The Fundamentalist’s critique of the Secularist:
The Secular view seems even more defeatist to me than the Mystic one… The Secularist doesn’t even want to bother searching for an underlying reality, since it simply doesn’t exist.
The Secularist’s Response:
The Fundamentalist needs to open his eyes to the world that lies outside theoretical physics. Of course there is a thrill in the thought that your research is opening up the blueprints of the universe, but it is the guilty pleasure of surrendering to vanity and parochialism… The whole approach is shamelessly ahistorical… Reality itself is not lying under anything. It is available for us to explore by scientific or other means…
The Mystic and Fundamentalist both want to go further, to an unimprovable understanding. They have some concept of a hidden universal truth. But this can only be defined as that which is yielded by the method they suggest for finding it, and there is no reason to think that this method will converge to a unique result.
The Mystic’s critique of the Secularist:
Even though the Secularist claims not to be a dualist, and affirms unity between the aspects of mind and matter, his description of mind and matter as two different “aspects” makes the two so separate as to make it very hard for me to see where the unity lies… In the Secular view, the two “aspects” that we “happen to distinguish sharply” form an effectively dualistic picture of the world.
The Secularist’s Response:
[negligible]
The Secularist’s critique of the Fundamentalist:
The Math-Matter link plays a crucial role in the Fundamentalist’s diagram [which lacks the Mind-Math link]. It corresponds to a very strong set of statements:
- Physics is converging to some set of ultimate laws,
- Our study of mathematics is converging towards an “ultimate math”.
- The ultimate laws will be expressed in terms of the ultimate math. Thus math has a special relationship to the material world.
- It is plausible (or perhaps just aesthetically appealing) that the ultimate math is not just an external reality, but that our Universe consists of it.
I find all these claims to be either meaningless or dubious… It is not clear that physics is converging to a set of ultimate laws… The theory that survives uncontradicted may do so mainly because the difficulty of falsifying it becomes insuperable… When this happens, the human need for novelty will lead science down other, more rewarding, paths than the search for a fundamental theory.
Mathematics has taken radical turns in the past and will do so in the future, stepping out in new directions that confound attempts to systematize it once and for all. I think this makes it impossible to formulate a coherent concept of the “ultimate mathematics”…
Nothing is gained by reformulating “it is dark in here” as “I have few visual sensations” or “the local intensity of electromagnetic radiation is low”. Talk of conscious beings as “self-aware substructures” is a similarly empty transcription.
The Fundamentalist describes himself as a formalist and a Platonist, but these are contradictory. A Platonist believes mathematical truths are truths about some world of mathematical objects, while a formalist believes that math is just the sum of all strings that you can get by manipulating symbols according to rules, starting with arbitrary axioms. As Gödel’s theorem shows, these are two different things: the methods allowed by formalists cannot prove all the theorems in a sufficiently powerful system. There are systems as powerful as arithmetic that are consistent and complete, and that therefore cannot be axiomatized, and so are outside the formalist structure. This spells doom to the Fundamentalist’s project. The idea that math is “out there” is incompatible with the idea that it consists of formal systems.
The Fundamentalist’s Response:
I would rephrase my assumptions as a Fundamentalist as follows:
- That the physical world… is a mathematical structure.
- That all mathematical structures exist “out there” in the same sense.
I believe that if we fail in this quest [finding the ultimate fundamental laws], it will be because of the limitations of our human minds rather than because of the nature of reality… I assume that there is an external reality that exists independently of us humans, and the laws of physics are how this reality works.
The notion of a mathematical structure is rigorously defined in any book on Model Theory. The integers are well-defined even though most of them have never been used in human calculations, and mathematical structures are likewise well-defined even though most of them have yet to be explored by mathematicians.
To me, all use of human language is necessarily loose and hence insufficient for describing an external reality existing independent of us. This is why nature speaks the language of mathematics and why I am advocating mathematical language to describe reality.
Abstract mathematics is so general that any fundamental “theory of everything” that is definable in purely formal terms (independent of vague human terminology) is also a mathematical structure. For instance, a TOE involving a set of different types of entities (denoted by words, say) and relations between them (denoted by additional words) is nothing but what mathematicians call a set-theoretical model, and one can generally find a formal system that it is a model of. In other words, if the physical world exists independently of us humans, it is not obvious that it can avoid being a mathematical structure.
To avoid [the] conclusion that mathematical and physical existence are equivalent, one would need to argue that our universe is somehow made of stuff perfectly described by a mathematical structure, but which also has other properties that are not described by it. However, this… implies either that it is isomorphic to a more complicated mathematical structure or that it is not mathematical at all. The latter would be make Karl Popper turn in his grave, since those additional bells and whistles that make the universe non-mathematical by definition have no observable effects whatsoever.
My hypothesis is that only Gödel-complete (fully decidable) mathematical structures have physical existence… If you define mathematical structures as (equivalence classes of) models of axiom systems, then they are guaranteed to be Gödel-complete (consistent). Please note that although we conventionally use a Gödel-undecidable mathematical structure (including integers with Peano’s recursion axiom, etc.) to model the physical world, it is not at all obvious that the actual mathematical structure describing our world is a Gödel-undecidable one… Even a world corresponding to a Gödel-complete mathematical structure could in principle contain observers capable of thinking about Gödel-incomplete mathematics, just as finite-state digital computers can prove certain theorems about Gödel-incomplete formal systems like Peano arithmetic.
The Mystic’s critique of the Fundamentalist:
I share the Secularist’s doubts about the idea that the world is “made of math”. My argument against this is that it seems to me like a category mistake. The category of existence of physical objects is different from the category of existence of mathematical objects.
The Fundamentalist’s Response:
[negligible]
