Everything has to be based on observation, provided observation is understood correctly. Observation includes grasp of essence, which is simply left out of what is called experience most of the time – in particular, by the empiricists.
Kurt Gödel [1]
This blog post was originally submitted to the FQXi Essay Contest 2020.
Introduction
This essay points to hidden premises in the arguments demonstrating the existence of mathematical propositions that are formally unprovable. It is argued that blindness to such premises has confused the interpretation of unprovability and uncomputability theorems. The same blindness has led many to assign improper and too prominent roles to codes and computation, roles they cannot play regardless these theorems.
Similarly, there are hidden premises in the rules we use to extract predictions from quantum theory. The source of the hidden premises is the same in the mathematical and the physical cases: we have to assume outside agents who act on the system being studied – be it a formalisation of arithmetic, or an experimental setup designed to test Bell’s inequalities.
The relation between the agent and the system has a specific form. Everything with specific form can be used as a handle to new insights. In this case the claimed common source allows us to find common traits of formal unprovability in mathematics and physical indeterminism.
To find these common traits, we need to open our eyes. We are blind to the nature of the relation between ourselves and the world since we are born with it, and thus take for granted – just like we are deaf to the harmony of the spheres since we always hear it, or so it is said.
In the beginning was time
There are some premises of reasoning we cannot do away with, but which we often ignore. One such premise is the existence of time. Ignorance of this premise sometimes leads to contradictions in terms. There are physicists who deny the fundamental nature of the directed flow of time. We may imagine one of them saying: “Yesterday I realised that there is no need for time in physics, today I’m writing my result up, and tomorrow I’ll explain it at a seminar.”
Time is a premise even in logic. The prefix in the very word premise is temporal in itself (which is also the prefix of prefix). The premise comes before the conclusion. And the steps in our reasoning leading to the conclusion are like the ticking of a clock. We cannot shuffle the axioms used in a mathematical proof with the intermediate lines and the theorem, without destroying the logic. The order and direction of the elements in the proof are fundamental.
Computer games
I see a similar blindness to the premises of the reasoning when I hear some people discuss coding, and the idea that the laws of nature are like a computer program – so that life becomes a simulation. The very concept of a code requires two things external to the code itself: the structure or message that is coded, and a key or dictionary that translates message to code and back. If we use the word object in a very general sense, we may say that as soon as we talk about codes, we presuppose the three objects in Figure 1. We also presuppose a set of relations between them. The key must be placed conceptually between the message and the code.
As a first observation, it becomes self-contradictory to try to describe the universe as a code, since then we presuppose the existence of objects outside the universe, which there cannot be, by definition.
This argument can be made more concrete. If we are materialists, we must assume that the code, the key and the message are imprinted in physical objects. A computer code is imprinted in computer hardware, and the key is ultimately imprinted in the brain of the programmer. The structure that is coded may be, for example, the atmosphere. Then the code corresponds to a climate model, and we are performing a climate simulation. If we analogously claim to be able to simulate the entire universe, and entertain the idea that the universe is equivalent to such a simulation, we clearly presuppose physical objects outside the universe, making the idea incoherent.
No object is an island
Let us turn to the relations between the three objects code, key, and message. They are also presupposed as soon as you speak about codes. These relations often lack direction, since the roles of the code and the message can sometimes be interchanged. They may, for example, correspond to two different languages. The key is then a dictionary. Such relations are implicitly presupposed whenever we speak about a set of objects. In fact, the very concept of a set implies a directed relation from the set to its elements, expressing the hierarchic relation between the object that corresponds to the set to the objects that correspond to its elements.
When we encode mathematical structures such as sets in a computer program, we sometimes forget that we also encode the relations between the elements in this structure. Look at matrices, for example. We encode the magnitude of the numerical elements, of course, but also their relations via the indices that specify the position of the elements in the matrix. The rows and columns are overlapping sets of elements. In the above figure, object C may correspond to column 1 of a 2 x 2-matrix, and objects A and B to elements e_12 and e_22, respectively.
It and bit
A computer code can be seen as a string of symbols that translates to a sequence of bits fed into the computer. Such a code is completely flat, lacking any hierarchical structure, and there are no inherent distinctions between objects and relations in the code itself. Therefore we might get the following idea: since we can encode and store in a computer memory a structure such as a matrix as a string of bits, hierarchies and relations among objects are not fundamental.
However, these hierarchies and relations are not grinded into nothing in the coding process – they just move to the key. They have to be there in order to regain the structure of the matrix when it is retrieved from the computer memory. And since the key is indispensable whenever we speak about codes, relations and hierarchies defend their fundamental role in any meaningful representation of reality.
This conclusion is reinforced by the simple observation above that the existence of a code implies a key and a message, with certain relations between them. These relations are shown in Figure 1, and constitute meta-relations as compared to the relations within the encoded message. It is tempting to try to grind all these relations and meta-relations to a structureless heap by creating a meta-code that translates the entire setup of code, key and message – and all the relations within this triad – to a string of bits. But it’s easy to see that we just end up in infinite regress: the meta-code inevitably gives rise to a meta-key and a meta-message, introducing new interesting relations. To me, these simple considerations show the incoherence of the idea it from bit, and its younger sibling it from qubit.
Again, let’s be a little bit more concrete. Currently, informational approaches to quantum mechanics are popular, where information is understood in its binary form. It is true that the simplest quantum mechanical systems have two possible states, and therefore may be called qubits. But it does not follow that all other quantum mechanical systems can described as collections of qubits. Indeed, this is not the case, since there are other fundamental quantum mechanical systems which have more than two possible states, such as quarks with their three possible colour charges.
Of course we may try to encode these more complex systems as a collection of qubits, but then we inevitably presuppose something external to the heap of qubits, as discussed above. And then the model is incomplete. In this case we need a key telling us which sets of qubits describe a truly two-state system, and which describe systems with more than two states. In other words, the qubits need to be marked, just like the elements of an encoded matrix need to be marked with indices telling us which rows and columns they belong to. Again, we may try to encode these markings into a meta-code, but we will just chasing our own tail in the never-ending dance of infinite regress.
The conceptual tower
The scientific ideal is to be able to explain as much as possible from as little as possible. This ideal can be taken too far, however. One lesson from the above considerations is, I think, that we cannot choose a too small or too simplistic set of assumptions and building blocks if we want to construct a complete model of the world. We must take all of its fundamental structures and relations into account.
The spatio-temporal relations between physical objects can hardly be reduced to something more fundamental, even if some physicists try. I argued that attempts to get rid of temporal relations lead to self-contradiction. Equally fundamental are the set-theoretical relations between physical objects that we most often take for granted.
We gladly speak of composite objects like a set of three quarks in a nucleon, a set of nucleons in an atomic nucleus, a nucleus and electrons in an atom, a set of atoms in a molecule, and so forth. Yet it is a profound state of affairs that perceived physical objects can always be arranged in this hierarchical, set-theoretical manner, a fact that makes the reductionist scientific method possible. These relations between physical objects define a proto-space of relative spatial scales. Kurt Gödel referred to set-theoretical relations as quasi-spatial [1].
The perceptions of temporal relations and the perception of some physical objects being parts of others are as fundamental to our perception of reality as the perception of the objects themselves. We may call them forms of perception (or forms of appearances, like Kant did [2]). To make more refined interpretations of what we see, to understand it and to build scientific models that account for it, we need still more advanced concepts, such as algebra and analysis. Therefore, there is a sense in which the entire tower of conceptual tools at our disposal has an equally fundamental position in our scientific worldview as the physical objects.
Kurt Gödel embraced this kind of conceptual realism. He claimed that we perceive concepts directly in a similar way as we perceive physical objects. He arrived at this Platonism at the age of nineteen, and said that it led him to his groundbreaking work in mathematical logic in his twenties. Naturally, he vehemently opposed the logical positivists of his day, who claimed that simple facts verified by empirical observation were a sufficient basis for philosophy and science.
Evidently, I agree with Gödel in this respect. We cannot derive simple concepts from facts, or all higher concepts from the lower ones. If we try to do it, we will have to use the concepts we try to derive, making the arguments circular. No conceptual bootstrapping is possible. If we just try to encode the concepts, we end up in infinite regress just like before.
Think again about the idea it from bit. It is indeed an idea. The idea that the world can be described as a binary code is very abstract, yet simple and indivisible. It cannot itself be decomposed in binary form. Then this idea does not belong to the world. Then what kind of world are we talking about?
Some self-reflection is needed
If we accept the fundamental nature of all elements in the tower of abstraction, all the way up from simple observations, then what kind of reality does this tower represent? Clearly, it cannot correspond to a naive materialistic world-view, in which the only things granted fundamental significance are physical objects or fields distributed in space and time, and where everything else can be explained in terms of these.
We need to find a common denominator for all levels in the tower, just like the common denominator for all physical objects is that they are located in the same three-dimensional space. To me, the only conceivable common denominator is that all levels in the tower are perceived or appreciated by someone; they are all integrated in the mind of an observer. In other words, all building blocks of the conceptual tower are located in the same mental space. This makes perceiving subjects equally fundamental as perceived objects. The relation between the observer and the observed becomes one of the fundamental relations of the world. This relation may be represented as follows.
When you think about this figure, it’s a bit paradoxical. In order to emphasize that subjects are equally fundamental as objects, we represent the subject as an object related to another object. You, the reading subject, look at this picture from the outside, and thus escapes the representation. We might try to represent this predicament like in the figure below.
However, you still look at this meta-representation from the outside. Clearly, whenever you want to catch the subject in order to incorporate it in an objective description, it always flies higher, escaping the butterfly net. If we repeatedly try to catch the subject in the objectification net we just end up in a similar kind of infinite regress as we already discussed in relation to messages and codes. The relation between subject and object also escapes this objectification net, emphasizing the fundamental nature of this relation. These reflections are examples of self-reference. The subject looks at itself from the outside as an object of study, and reflects on its own nature, and its relation to the objects it perceives. In such a case we may divide the single bi-directional relation between subject and object in Figure 4 into two directed relations that form a loop. We may read the upper relation as: The subject looks at an object, and the lower relation as: This object represents the subject.
In a more concrete setting, the representation of the subject may simply be our mirror image. Generally speaking, self-reference does not require a self in the sense of a personal subject, but just two or more objects with directed relations that form a loop. We may take acoustic feedback as an example, where the two objects that are related in a loop are the microphone and the loudspeaker. We need at least two objects in such a loop of self-reference; a single object referring to itself makes no sense, I think. We need the mirror image to see ourselves; I cannot see my face directly with my own eyes.
Semantic self-references may seem genuinely self-referring, but this is not really so. Consider the famous liar paradox: This statement is false. This sentence is actually the set of two parts that carry meaning: a subject and a predicate, where the predicate refers to the subject.
A: This statement (subject)
B: is false (predicate)
C: This statement is false (sentence)
The first part A does not refer to itself, but to C. Referring to Figure 2 we may draw the following figure. Here we have two loops: one with two objects A and C, and another one with three objects A, C and B.
With the liar paradox we approach Gödel’s famous first theorem about the incompleteness of axiomatic arithmetic.
Looking at a theorem
Let me try to illuminate the structure of Gödel’s proof in the light of the discussion so far. Much more detailed – but still generally accessible – accounts can be found in several places, for example in the charming little book What is mathematical logic? [3]
The basic idea is just to modify the Liar paradox in the following way.
A: This statement (subject)
B: cannot be proved (predicate)
C: This statement cannot be proved (sentence)
This statement is self-referential in the same way as the Liar paradox, as expressed in Figure 8. Clearly, it must be true to avoid contradiction. Thus we have a true statement that cannot be proved, and thus truth goes beyond proof.
This is just cheap semantics, of course. To make a precise mathematical theorem out of it, we first have to replace the word statement with formula. Further, we must refer to a specific formula, which we may call F. Since the word statement defines part A above, but also refers to the entire sentence C, the entire claim This statement cannot be proved must itself correspond to formula F. To create such a correspondence we must find means to code sentences such as C as formulas of the same type as F. We may then write
A: Formula F
B: cannot be proved
C: This statement can be coded as a formula F, which cannot be proved.
The above formulation makes the hierarchical, set-theoretical difference between A and C explicit: whereas A is a formula, C is a statement about a formula (which in turn can be coded as a formula).
The formulas and proofs we are talking about here belong to the realm of formal systems. Below is shown an alphabet of 13 symbols that is sufficient to express any arithmetic proposition as a string of these symbols called a formula. For example, the number two is expressed as s(s(0)),where the symbol s can be interpreted as the successor of. The number two is clearly the successor of the successor of zero.
To a formal system also belongs a set of axioms and a set of allowed inferences. The axioms are just a set of formulas, and the inferences are mechanical rules that allow us the get new formulas from an old one. A proof of a formula F becomes a sequence of formulas, where the first one is an axiom, and the last one is F.
To code a statement about formulas and the possibility to prove them into a formula, as required in sentence C, Gödel introduced a method to code formulas and proofs as integers, now called Gödel numbers. One way to do it is shown above. It uses the uniqueness of the decomposition of an integer into a product of primes.
The coding makes it possible to say whether a given integer is the Gödel number of a formula, or a proof, or neither. If it is the Gödel number of a formula, the formula can be decoded, and if it is the Gödel number of a proof, the proof can be decoded.
With the help of the coding schemes expressed in Figures 9 and 10 we can close the semantic loop in Figure 8 in a mathematically well-defined sense.
We have to find a formula F that closes this loop and makes it self-consistent. Such a formula is not too hard to formulate [4]. As a consequence we get Gödel’s first incompleteness theorem: In any consistent formal system with an interpretation that contains arithmetic, there is a formula F that cannot be proved or disproved within the system. Since the arithmetic interpretation of any formula is either true or false, it follows that in any formalisation of arithmetic, there is a true proposition that cannot be formally proved. Here truth is defined within element D in Figure 11, whereas provability is defined within element C.
A common but controversial inference is that human reasoning is something else than the mechanical procedures of formal proofs and the execution of computer programs. We seem to have access to more truth than a computer can ever have, and therefore we must stand outside and above the mechanics.
What I want the reader of this essay to take to heart is the following: we make use of the last statement in order to prove Gödel’s theorem. It is a presupposition. The inference is therefore a tautology. (Gödel’s theorem itself is certainly not.)
If we want the proof of Gödel’s theorem to stay within the formal system it analyses, the formula F must speak about itself. But there is no genuine self-reference in the sense of Figure 7. Instead, to make the argument in the proof go through, we presuppose three elements in a loop, where the formal system under study is just one of them. We also use the hierarchical nature of set theory according to Figure 2. Such hierarchies are non-existent within a formal system.
Hey you
Elements C and D in the loop in Figure 12 correspond to propositions and statements. They carry meaning, which must be defined subjectively. What we presuppose in the proof of Gödel’s theorem is therefore an agent who observes the formal system from the outside. We presuppose ourselves, naturally.
In this picture, the arrows pointing to and from A in Figure 12 correspond to a relation between the observer and the observed. In his proof, Gödel demonstrates that this relation is fundamental in order to understand mathematics, and the limitations of its formalisation.
There is an analogous situation in physics. It was realised in the beginning of the twentieth century that there is a limitation to classical, Newtonian physics: it cannot account for all observed phenomena. Quantum theory was therefore invented. A decisive difference between classical physics and quantum theory is that the latter treats the relation between the observer and the observed as fundamental. This is so since we find a rule among its postulates that tells us which values of a quantity can actually be observed in a given situation, and also a rule that gives us the probability to observe each of the allowed values.
Many physicists want to explain away this fact, trying to derive these postulates from the others. What they want to do, in effect, is to objectify the act of observation. They argue that as far as physics concerns, it is sufficient to consider ‘observations’ made by a measuring apparatus. Such an ‘observation’ just amounts to an interaction between the system of interest and the apparatus. I have argued in relation to Figure 5 that it is logically impossible to erase the subject in this way; we just end up in infinite regress. The messiness and incompleteness of the mathematical frameworks that come out of these attempts suggest that it is a dead end in physics as well.
Self-dissection
At the same time, the truly materialistic approach to the physical world has proven immensely fruitful in the past. And neuroscientists have unravelled close neural correlates to our subjective mental states. How do these facts go together with all of the above?
We may try to make all these things fit together by introducing an operational version of materialism, in which each shift in our subjective perception corresponds to a shift in the brain state. The odd thing in our predicament is that the brain state is a state we observe according to the laws of quantum theory, just like all the other things around us. The brain state is therefore itself a subjective perception, in a sense. We get a self-referential loop in physics similar to the one in mathematics that Gödel pointed out.
We have to pause for a minute, though. If operational materialism is interpreted very strictly, it should be possible to explain each of our actions as a result of the processes going on in the brain. There are two potential problems with this picture – one mathematical and one physical.
Let us start with the mathematical problem. In this picture, the brain should be seen as analogous to a computer. But then it could also be seen as analogous to a formal system, mechanically processing input like axioms, producing output like true formulas. However, we have argued that the structure of Gödel’s proof presupposes something fundamental that is external to the formal system itself, yet interrelated to it. There is no place for such subjective externalities in this strict version of operational materialism.
The physical problem has to do with causality. Quantum theory is expressed in the following form: Given that we observe a certain system in a certain way, the probability to see this or that is so and so. If I look to my right I will see this or that with certain probabilities, and if I look to my left I will see this or that with other probabilities. But the theory does not say anything at all about which direction I will actually look. This is often called Heisenberg’s choice, and it lies outside the scientific description, as far as quantum theory can tell. There are not even any probabilities associated with it.
Heisenberg’s choice is fundamental within the theory and has tangible consequences; it affects the future evolution of the physical world. Yet it is impossible to account for it in a picture where all choices reflect the actions of a brain whose state evolves according to the laws of physics. This is true even if those laws are probabilistic, like in quantum theory. (We may, of course, try to express Heisenberg’s choice as the outcome of a probabilistic quantum experiment, but we just run into the arms of our old friend infinite regress.)
To make the idea of operational materialism consistent with known mathematics and physics we clearly have to find its limitations. Again, there is one mathematical and one physical side of the coin.
Mathematically, or rather logically, we may construct a self-referential loop where a subject observes her own brain state F as she is about to make a choice. The choice may then be such that it contradicts the state F supposed to give rise to it. The set-up is similar to the circuit shown in Figure 11 in order to motivate Gödel’s theorem.
In this case the conclusion would be that there are brain states F such that a choice is made by the owner of the brain that cannot be accounted for by F.
We may try to find the physical source of the limitation to operational materialism in the architecture of the brain. We should clearly search for a microscopic structure that is crucial for information processing, having the potential to influence macroscopic action. It must further be able to hide its inner state from outside scrutiny, at the same time as this inner state may influence the information processing being performed by the structure.
The so called microtubules are intriguing candidates. These extremely thin tubes are abundant inside neurons. Their circular circumference is composed of just thirteen tubulin molecules, whereas their length can stretch to the centimetre scale. The tubulin molecule has a discrete set of possible conformations. Conformational changes can spread like waves along the tube, suggesting that the surface of the microtubule acts as a cellular automaton, performing computations relevant to brain function.
The interior of the microtubule is filled with water. One may speculate that the state of this water is unknowable to a large extent. This may be so because the rigid lattice formed by the tubulin molecules only allows a discrete number of collective states of the entire microtubule surface, whereas the possible states of the water inside the tube is probably much higher. If this is the case, some information about the inside of the microtubule is necessarily lost at the surface.
In this way we may be able to avoid the possibility to gain perfect knowledge of our own brain state. Such a limitation would make the idea of operational materialism consistent with the limitations of computability and determinism highlighted by Gödel’s theorem and quantum theory, respectively. It would also make operational materialism conform with the fundamental role given in this connection to the observer and her relation to the observed world. I come to think of a quote by Niels Bohr:
The idea suggests itself that the minimal freedom we must allow the organism will be just large enough to permit it, so to say, to hide its ultimate secrets from us. [5]
Stuart Hameroff and Roger Penrose have argued forcefully for the idea that microtubules play a crucial role for the appearance of consciousness. [6] Their perspective is somewhat different from the one I present here, however. Several recent studies [7] point to the important role of microtubules for the higher brain functions, even though the underlying mechanisms are still unknown.
The conceptual clock tower universe
The world view promoted in this essay incorporates the observing subject in a never-ending dance with the observed world. The couple is forever interconnected by a fundamental relation. It also incorporates fundamental forms of perception such as time and set-theoretical hierarchies into which the naked sense impressions are placed, and also some more abstract concepts used to interpret and relate them. It is claimed that such a world view is often implicitly presupposed even by those who explicitly deny it.
Sense impressions, forms of perception, and higher concepts are seen as equally fundamental in this world view, and define different floors of a conceptual tower. It is suggested that the world is this tower, with all its inhabitants and furniture. Then the outline of the tower, together with the stairs that run between the floors, the pipes and ventilation shafts, must correspond to the laws of physics, as well as to the structure of the mathematics used to express these laws. If this is true, forms of perceptions and the structure of mathematical concepts should give clues to the laws of physics, and vice versa.
Since the perception of physical objects in three-dimensional space is just one of the floors in the tower, the tower itself is not built in this concrete, ordinary space, but is located in a more abstract space, which cannot be separated from awareness.
The objectivity of such a world does not reside in the physical objects that are observed in three-dimensional space. Rather, it resides in the parts of the conceptual tower and how they fit together. The tower has its rotating cogwheels, like a mental mill, or a clock. True, this is another kind of clockwork universe than that suggested in the seventeenth century. Even so, it is not for us to decide how the conceptual clock tower operates in its abstract space, what the pieces look like and how they fit together. That cannot be changed by our wishes. We can still get crushed between its cogwheels. In that sense such a universe is still objective.
References and notes
[1] Hao Wang, A logical journey: From philosophy to Gödel, The MIT Press (1996).
[2] Immanuel Kant, Critique of pure reason, translated by P. Guyer, and A. W. Wood, Cambridge University Press (1998).
[3] John N. Crossley et al., What is mathematical logic? Oxford University Press (1972).
[4] Since element C in this loop is a statement about a formula whose nature can be varied, we should consider corresponding formulas F’(x) with a free variable x. We may then specify F as follows, as an arithmetic proposition within element D in the loop: There is no y such that y is the Gödel number of the proof of formula F with Gödel number z, where F is defined as F’(x) with x replaced by z, as expressed symbolically in the formal alphabet.
[5] Niels Bohr, Light and life, Nature 131, 421-23 (1933).
[6] Stuart Hameroff, and Roger Penrose, Consciousness in the universe: a review of the ‘Orch OR’ theory, Physics of Life Reviews 11, 39-78 (2014). [7] Travis J. A. Craddock et al., Anesthetic Alterations of Collective Terahertz Oscillations in Tubulin Correlate with Clinical Potency: Implications for Anesthetic Action and Post-Operative Cognitive Dysfunction, Scientific Reports 7, 9877 (2017); María del Rocío Cantero et al., Bundles of Brain Microtubules Generate Electrical Oscillations, Scientific Reports 8, 11899 (2018).
