We do not understand information. Despite living in an "information age" of all-pervasive information and communications technology (ICT), we have several radically different and contradictory models of the stuff, and in consequence we run into all sorts of paradoxes when we try to theorise about it. I have this bright idea than the mathematicl field of topology has the potential to unify these disparate models and give us a firmer foundation of understanding, at least for the physicist and perhaps also for the technologist and the philosopher.

But first, perhaps I need to remind you of some of those different models and their inconsistencies. There is a discipline called information science, which tries to join the dots between the way people and computers work with information. Another is called information theory, and deals with such things as packing more data onto a radio channel or correcting errors in a noisy one. Information theory links the stuff closely to the thermodynamic concept of entropy; information has entropy. Cosmological black holes make the link even more explicit; both the entropy of a black hole and the amount of information it has swallowed are directly proportional to the area of its event horizon, the boundary from within which nothing can escape. In thermodynamics, entropy never decreases but can and usually does increase. But quantum theory has its own idea of information, in which information is never created or destroyed, it can only ever be transformed. Note that both thermodynamic and quantum information are factual, in that they describe the physics. They are always true. Where errors do creep in, they are caused by other, extraneous information, called "noise", creeping into a measurement. By contrast, the primary domain of information science is the symbolic or semantic information â€“ what the chatter over those physical channels actually means. This stuff can be, and often is, false, whether by accident or design. Even keeping philosophy out of it for now, that's three different and contradictory kinds of information and counting; thermodynamic, quantum, and semantic.

Confused yet? You should at least be going that way. The rest of this essay is my attempt to pull all this together into a coherent picture for the physicist.

To deal with physical information first; that is, as described by theoretical physics in the form of thermodynamic and quantum information. How come these two descriptions are so different? The one tends always to increase, the other cannot. I would suggest that these can be reconciled, once we realise that thermodynamics is time-bound but quantum mechanics is not. The third pillar of modern physics is relativity. It famously describes all of spacetime as a fixed "block universe", across which timelines or tracks trace out our individual journeys. It will help us visualise the resolution to the paradox.

Thermodynamics is essentially time-bound. Its Second Law, that entropy always increases, embodies a causal arrow. That arrow flows forward in time, creating "time's arrow" and making clocks run forwards not back. If you look at your timeline and find "now" on it, thermodynamics describes the universe from this perspective, telling you about the changes you will experience as you travel along your track.

Quantum mechanics is not time-bound in the same way. Firstly, there is its mathematical foundation in the "sum-over-histories" approach. The wave aspect of any quatum pervades not just its local future but all of spacetime. When you measure it, it takes every possible path through spacetime to get to the measurement point. These paths all interfere with each other, leaving just the standard wave equation of actual "possibilities". You might think this picture to be, as a minimum, utterly insane, except that it predicts a limited and subtle breaking of time's arrow, known as nonlocality. This nonlocality raises its head in many forms, and has been established experimentally beyond doubt. Only when you have considered the entire past, present and future of the whole universe can you say what your time track will probably be. Thus, quantum physics tells you about the state of your whole track and has nothing to say about which "now" point you want to look at.

I would suggest that the different treatments of time in thermodynamic and quantum approaches are due to the nature of the observer; are you a 3D object trapped at some point in time, or are you a 4D or even 5D superbeing gazing down on the entirety of spacetime? Such changes in viewpoint, even when they involve the number of dimensions, may be described using mathematical device known as a transform or transformation. A particular transform of interest here is that which underlies twistor theory. Here, we transform our view from the spacetime of relativity to something known as twistor space, in which the entire time track of a photon becomes a point, and a point in spacetime becomes smeared out across every photon which passes through it. Twistor space has no concept of time passing. It also turns out to be the natural home of quantum interactions, condensing perhaps twenty pages of dense calculations in relativistic spacetime onto a single page in twistor space. Thus, the apparent paradox of conflicting relationships between time and information turns out to be one of viewpoint. In quantum physics, an object and the information about it exist across all time, there is no way in which that sum of information can change from moment to moment. The quantum information is just the sum of all those timebound moments of information. While the local information is different from moment to moment, its total sum along the timeline is fixed.

Now I want to turn to the third type of information, embodying symbolic meaning.

Symbolic meaning or semantic information means anything only to its observer. For example if you drag a few atoms over the surface of a crystal, they will make a pattern of some kind. A researcher once dragged a couple of dozen or so to spell out the name of their company, IBM. He had added information that was not there in the physics; it had moved from the realm of information theory to that of information science. Or, take the verb "demand". It means rather different things to a Frenchman and an Englishman, an issue which has caused more than one diplomatic misunderstanding in the past. Visually, the symbol "+" may take a wide range of meanings depending on context. In other words, semantic information is heavily dependent on context, in a way that physical information is not.

Another property of physical information is that it is always true. It is simply a property of the physical entity. Neither entropy nor the quantum ever lies. Data errors are introduced only where some external physical information, termed noise, gets picked up. By contrast semantic information is often false, sometimes deliberately so, or even in some limbo state such as paradoxical or wholly meaningless.

Physical information cannot be regarded as a subset of semantic information, say the "true" subset. Yes, we can create a parallel semantic copy of the physical information of interest. But this semantic knowledge is in essence a property of the observer and not of the physical symbol or entity carrying it. This is true of all semantic information. Knowing what the Rosetta stone says changes the archaeologist, it does not change the actual stone in any way. Famously, left to themselves black holes evaporate. As the event horizon shrinks, it re-emits all that physical information, albeit radically transformed. A spaceship's log book might be swallowed, and the physical information about its data store eventually transformed and re-emitted as Hawking radiation. But any semantic information in it, such as what was the last entry made, is permanently destroyed. Thus, we can state categorically that semantic information is quite distinct from physical information.

This strikes an obvious parallel with the role of the observer in quantum physics. Here, the quantum state of some entity exists as a range of possibilities until it is observed; its state then "collapses" into that observed. This can be understood as the moment of copying from a physical to a semantic state. [But what if the "observer" is something lacking intellect, such as a wider entity which the quantum under consideration has just banged into? SchrÃ´dinger's cat, for example. Some argue that this whole setup is now in an indeterminate quantum state, and so on until a conscious observer finally steps in; a paradox known as "Wigner's friend". Others disagree profoundly. I need to think about this.]

This section is at present more just a statement of intent.

Topology, or "rubber-sheet geometry", is the branch of geometry which describes the shapes of things in a general kind of way; how many holes, how many twists, how many edges? Any kind of surface or space is a topological manifold. Examples include relativistic spacetime and the Calabi-Yau manifolds in string theory. It is therefore much beloved of theoretical physicists studying these things, however my purpose in raising it here is a bit more abstract. I want to introduce some of its concepts as a way to think about information, and especially the nature of quantum measurement.

I want to distinguish two kinds of geometrical operation. The mathematical terminology can be a little loose and/or confusing, so please bear with me. If one object can be smoothly transformed into another without any cutting or gluing, we say that they are isomorphs. I am going to call such an operation a morphing. A simple example is offered by perspective drawing, where an object is distorted by being seen from a short distance away. It may be distorted, but it still retains the same structure, the same morphology. But topology also deals in more drastic operations. Suppose we cut a hole in the side of a plastic cup and glue one end of a plastic straw over the hole. We now have a jug, with two holes through it. I will call this surgery. Both kinds of operation are powerful tools of topological analysis.

There is another kind of change in viewpoint, where the mathematical expression of the space in which an object exists is changed, but the object itself is not. These changes are sometimes called transforms (as distinct from transformations), and I shall stick to that. An everyday example is whether we choose to treat a complex sound as a wave in space and time (as loudspeakers do), or as a frequency spectrum (as sound equalizers do). Transforms are powerful things, they are apt to make whole dimensions appear or disappear, or change their properties in fundamental ways. A hologram is an example of this, where a small solid object in three dimensions gets smeared out across the whole of a two-dimensional space. The object appears to change dramatically, but in fact it is just the same object seen through different mathematical eyes.

I want to suggest two ways in which all this relates to information:

- The change in viewpoint between thermodynamic and quantum descriptions of information is a transform, in which the information is the same but our understanding of it is dependent on our mathematical formulation. Recall that such transforms can mess around with dimensions, and it is this messing around which makes information behave differently in the eyes of the thermodynamicist and the quantum physicist.
- Reading "IBM" into a pile of atoms was a major surgery; the morphology of the physics was abandoned in favour of that of the observer's memories. This illustrates that the transition from physical to semantic information is by its nature a surgical one.

Updated 14 May 2023