Beyond Integrated Information Theory

Steven Schkolne
14 min readNov 13, 2020

A bold theory has seized the scientific community. Despite being tragically incorrect — and receiving over a decade of considerate evaluation — it won’t go away. While IIT badly misses its target, it leaves us with lessons that are vital to moving the discussion of consciousness forward.

Called Integrated Information Theory (IIT), this framework aims to explain consciousness in physical terms. IIT is in a competition to become the established scientific view of the phenomenon. Boldly, IIT claims to measure that ineffable thing called consciousness with a single number, 𝚽. If their theory is true, your dimensions not only include height and weight, but also your consciousness value.

IIT does not actually fit any data and is made of opaque, custom math. This makes IIT difficult to refute. You can’t point to contradictory evidence because the theory cannot, in practice, be falsified (applying it to human consciousness is impossible). Few can argue against the math because, along with being rather advanced, IIT uses many custom constructs.

IIT also lingers because of several strengths of the theory, which are not inconsequential.

This leaves the world in a weird place, a spot that bothers me. I encountered IIT in my research on machine consciousness but never took it seriously. Chatting with elite neuroscientists about my project, they kept asking me about IIT. What did I think of it? Whenever I turned the question around, I got a curious answer: “I don’t quite get the math”. How can we, as a community, make sense of the evolving scientific landscape around consciousness if we can’t accurately place IIT? Is IIT complex out of necessity, or as a survival mechanism? (to stop ordinary, non-IIT-obsessed neuroscientists from swatting it down)

I took the time to dive into every nook and cranny of the IIT formulation. My initial goal, to create an explanatory IIT resource, changed as I realized the payoff was not worth the effort for most readers. Even a streamlined explanation would take hours to read, and require no shortage of mental effort. When I realized the greatness of these depths, along with several grave errors in the theory’s construction, I switched tactics.

Instead I’ll lay bare for you several flaws I see in the theory, constructively framed as “Lessons” to be learned for the next round of work. I’ll also document the terrific contributions IIT has made towards unravelling consciousness (“Foundations” for future study). You need to make up your own mind about IIT, I hope this article furthers those ends. For those who will dive deeper by reading the IIT paper and muscling through the math, I hope my tips speed your journey.

Lesson #1: Don’t Stray From Evidence

Models with predictive power are the essence of science. When we look at Newton’s F=ma, it does something very basic but quite reliable: force can be predicted accurately from mass and acceleration. Newton used observations to both generate his famous equations, and also to prove them correct. While IIT is inspired by evidence, its equations do not actually fit any observational data.

IIT’s equivalent of Newton’s F is the consciousness value 𝚽. Yet 𝚽 has never been measured for an actual brain. Such a feat can’t be accomplished today, and may never be possible. For starters, calculating 𝚽 requires measuring the internal state of every neuron in the brain at once. We are far from measuring the brain at that resolution, and that is just the first obstacle. A single measurement would not suffice, we would need to take a range of measurements over time to build a model with the requisite ability to predict the state of each neighboring neuron quite accurately. How would we get these probabilities? [1]

Even if we could take such measurements, IIT demands that we gather them for each of the 10²⁵⁸⁰⁰⁰⁰⁰⁰⁰⁰ possible groups of neurons (partitions in a power set [2]). Yes, that’s a 26-billion-digit number. Even in the world of big numbers, that’s a shocker. The estimated number of atoms in the universe is a mere 82-digit number. The age of the universe in seconds, an 18 digit number. Where would we find the time or space to sort all 10²⁵⁸⁰⁰⁰⁰⁰⁰⁰⁰ values to find the largest one? [3]

The authors acknowledge these shortcomings, but proceed despite them. They admit in their key paper that IIT is infeasible for any system having more than 12 neurons. Acknowledging this up front does not make the shortcoming any the less critical. The whole point of IIT is to measure human consciousness. What good is a brain-measuring ruler that can’t measure the only brain that matters?

IIT’s ruler can duly be applied to non-biological systems, and the results are damning. Aaronson’s Vandermonde system achieves a high 𝚽 score by evaluating complex polynomials, the homework of an Algebra II student at massive scale. By growing the size of a matrix, the Vandermonde system can reach arbitrary heights of 𝚽. This means that, no matter how high a 𝚽-value we may measure one day in a biological brain, a simple system (that lacks typical characteristics of consciousness — awareness, theory of mind, etc) can be made to have a yet higher one. When you trumpet a theory without first asserting its predictive power, prepare to be surprised!

The lesson here is to keep theory close to data. The Perturbational Complexity Index (PCI), which is heralded as providing experimental back-up for IIT, actually uses the Lempel-Ziv algorithm to calculate complexity. Yes, the dead-simple math that’s used to ZIP files for faster download powers the IIT crowd’s most heralded evidence of their measure. Lempel-Ziv is not special — any variety of Shannon information will do.

The PCI authors talk about “integration” in a way that is unrelated to the math of IIT. Instead of trying to shoehorn this data into IIT, we should start clean and ask ourselves: what math fits the data best? This is our best hope of developing an actual science of integration.

Lesson #2: Don’t Traffic in Axioms

Instead of data, the IIT authors rely on “self-evident” axioms to both derive and prove their theory. I was shocked when I encountered this — was this a joke? Unfortunately it is not.

It seems the authors — trained scientists — felt they were being academically correct, as philosophers, by starting with axioms. They cite Descartes as a precedent of this practice, perhaps clumsily revealing that their thinking is about 400 years out of date. As in science, the axiom has disappeared from philosophy, cultural theory, and psychology in favor of evidence and reason.

In today’s global society, nothing is self-evident, especially when it comes to consciousness. If you’re wondering, IIT’s axioms aren’t very convincing. They left me shaking my fist in wild disagreement. How can they claim consciousness is “undeniable”, have they not read Dennett? How can they assert “at one given time there is only one experience” when I can drive and listen to music at the same time. That seems to me like two simultaneous experiences (an introspection that is somewhat backed up by brain data). You may not agree with me, and that’s my point. The IIT axioms are subjective.

We can forgive the IIT authors for being lulled by the false comfort of indisputable axioms. Consciousness is a slippery subject, with a long track record of authors’ imaginings becoming so solidified in their minds that they mistake them for truths. We can understand this, but we should not accept it.

From their axioms, IIT’s authors derive a series of postulates, and from those postulates they build their math. That they can make math that satisfies their axioms is not surprising. What is disturbing is that they (naively?) believe that their math is the only formulation that works. Myriad other forms of math also satisfy these axioms, why are they not the unique measure of consciousness?

The only argument for the correctness of IIT is that it satisfies the authors’ preferred axioms. Given the shaky nature of axiomatic footing, this is glaringly insufficient.

This story is perhaps best viewed as a deep lesson about disciplines and expertise. We see in IIT some world-class neuroscientists demonstrating to us that, as philosophers, they don’t even raise to the level of your average undergraduate. If you’re a scientist, use data. Don’t base your scientific theory on a style of argument that is outside of your field of expertise.

Lesson #3: Don’t Fake Your Way to the Essential

There is a massive disproportionality between the simplicity of IIT’s axioms and the complexity of their math. If not from data, where do these intricacies originate?

The chase is fueled by the grand pursuit of a single number called 𝚽 which measures consciousness. One of IIT’s axioms is that of exclusion: there is always exactly one indivisible thing called “consciousness” in the brain at all times. 𝚽 is a measurement of that thing. The fact that they can measure something “exclusive” to produce such a number with their equations is like a proof of their axiom of exclusion.

But when we look under the hood at the methods they use, the specialness of this measure grows more and more difficult to believe. Exclusivity is found by measuring the integrated information of all 10²⁵⁸⁰⁰⁰⁰⁰⁰⁰⁰ groups in our power set. For each group of neurons, IIT measures how much integrated information is inevitably lost when it is split up. The mechanism-in-a-state that loses the most is considered the unique, exclusive conscious experience of the system.

While this formulation does get to something exclusive, it does so in a lazy way. It is like measuring a forest by the height differential of the tallest, and second-tallest tree. Sure, that is a single number that describes the forest. But what is really being described? Why use this number instead of the absolute height of the tallest tree? Or average the heights of the trees to determine the height of the forest?

Of all the various ways to come up with a unique number, IIT does something curious [4] to find their supposedly exclusive aspect of mind: they discard a whole bunch of activity. This is not a deep, fundamental discovery, but rather a mathematical hack. If IIT’s math was less opaque, perhaps more people would see it for what it is.

One lesson we can learn is to keep it simple. The opacity of IIT’s math bities itself in the ass. If your formulation is clear, then you can’t get away with stuff like this without being called out on it readily. Perhaps the old adage — if you can’t explain it to a non-expert, you don’t understand it — does require adherence. Do the IIT authors themselves see their error?

This leads to an additional lesson — don’t get romanced by the exclusive. Just because a mathematical process results in the computation of a single number, doesn’t mean that number is essential. Just because a measurement can be made, does not mean it is the measurement.

Lesson #4: Actually explain consciousness

Despite its complexity, IIT doesn’t begin to approach the pressing questions of consciousness: what is it like to be something? Are my qualia the same as yours? What is the nature of self and will? What are experiences for? Even authors who are accused of “not explaining consciousness” deal with these subjects.

Instead of explaining “concepts” and “qualia”, IIT redefines them. A concept is a mechanism-in-a-state: a flicker of activity in a system, neurons lit up in the brain. What does a concept of fish look like? What is the shape of democracy or truth or Mickey Mouse? Is our common notion of a concept the same as IIT’s technical term? The answer is unclear. Similarly with qualia. IIT’s qualia is the “shape” of their mathematical structure. Just because it’s a shape of a “concept” doesn’t make IIT’s “qualia” any more real (and by defining qualia as a mathematical structure they dodge the core argument of qualia theory, which is that there is something about the mind that can never be quantified).

It is fine to create new definitions and terms. Just about every author on consciousness does it. But the burden of consciousness theory is to answer specific topics. If your theory doesn’t engage the core questions, it doesn’t actually address the subject.

Any pursuit of IIT quickly diverges from a consideration of common consciousness questions into what feels like a fun house. IIT is a wildly creative theory: all kinds of things are simply made up. Instead of a platform for generating answers, IIT provides deeper layers of mystery. A MICS is generated by a complex, which generates integrated conceptual information, which requires a cause-effect repertoire, and before you know it you find yourself far from what you know consciousness to be. Any attempt to unravel IIT leads to a self-reflexive discussion of the theory itself, with no regard for anything external to the theory. The guts of IIT are made of fantasy stuff.

Moving Forward

Despite its weaknesses, IIT has some core strengths. Let’s not throw the baby out with the bath water.

Foundation #1: Mind Is State

The mind/body problem has been with us for millenia. IIT’s very formulation proposes a resolution of this gnarly tangle with the concept of state.

IIT talks about a “mechanism in a state”. It is in the “state” of the mechanism that they find consciousness: state (in the IIT sense) is mind (in the Cartesian sense). State is something all scientists are familiar with, computer scientists like me are always manipulating data: the dynamic state of machines. A chemist characterizes a gas with state variables: temperature, pressure, volume. A mechanical engineer talks about the state of a car: the angle of the steering wheel, the torque of the engine, the temperature of the radiator, etc (this list can get quite long!). State is the language we use to talk about the temporal status of elements in the grand universal simulation we call physics.

IIT is not unlike other scientific theories of consciousness. All the theories I know about, from Edelman’s reentry theory to the recent Attention Schema Theory, are based on state. Curiously IIT is the only theory to get a reputation for being “pantheist”, and making the assertion that everything is conscious.

Surprisingly, IIT does nothing of the sort. Very few things are considered conscious by the theory. Some quite complex mechanisms (feed-forward circuits) are not at all conscious (𝚽 = 0), a fact the authors are quite proud of. Basic digital circuits however are conscious. Remember the foremost goal of IIT is to distinguish conscious from not-conscious. Perhaps it is the crossing of the biological divide that makes people react so strongly to the “pantheist” nature of IIT (a phenomenon that I’ve attributed to human exceptionalism).

IIT sets a high bar for a theory of consciousness: information-state is assessed in a way that can be performed on biological and non-biological systems alike. Any competing scientific theories now need to meet a similar standard to be considered complete. Compared to IIT, other theories of consciousness now look weak because they only describe the human example of the phenomenon.

Foundation #2: Cause-Effect Information

Confusingly, IIT rolls with its own definition of “information” that matches neither our common sense of the term, nor the technical sense ushered in by Shannon at the dawn of the information age. By “information”, IIT does not mean the classical information of the mechanism under study, but rather “cause-effect information” which is a measure of causal entanglement.

The metaphor of a wire helped me build intuition for cause-effect information. Classical information theory is all about wires. We can send, say an image, over the wire, and see how much of the original information was transferred. Such a simple wire could roughly model the memory of a brain or computer: images go in, they are encoded, then they can be recalled.

Shannon’s information theory concerns such dumb wires, describing the likeliness that the information going in comes out the other side. IIT’s integrated information clicked for me when I envisioned it measuring a smart wire. Like our dumb wire, there is input (the past) and output (the future). Instead of merely shepherding information, this wire is a mechanism with its own state (classical Shannon information) that is entangled with the outside world. All kinds of things can happen.

IIT measures the causal entanglement of my smart wire, with a core focus on the inside of the wire, the state of the “mechanism in a state” that is being evaluated for consciousness. How predictive is what’s going on inside the wire of the past and future? If the answer is highly predictive, then there is a lot of cause-effect information [5].

IIT’s consciousness measure is, at heart, a measure of the causal entanglement of internal state with past and future state. Analyzing systems in this way is curious, I know of no similar precedent, and I find myself thinking about not only mind but other systems. IIT goes well beyond my simple wire example, their measure applies to any number of inputs and outputs, as well as feedback within the mechanism. It would be curious to measure the cause-effect information within all kinds of systems, and see what the measure indicates.

IIT’s best chance of being saved as a core theory is to start to build an actual foundation around this new measure of causal entanglement. A stripped-down version of IIT, anchored in nothing but cause-effect information, applied to actual data might reveal a domain where the theory has predictive power. A scientific theory can run on philosophical fumes for only so long!

The Path Forward

The consciousness community can take the strengths of IIT, and learn from its mistakes. Consciousness is an unflappably romantic subject, we can all easily become entranced with the idea that our particular vision of it is truth. If that were not the case, there would not be so many opinions about consciousness.

In a scientific context, IIT tries to make a theory based on “self-evident” axioms work. We see that just a few steps down this road leads to a framework that’s almost entirely religious. There is simply no data. Theoretical physicists do not veer so far from experiment, and neither should neuroscientists. It’s extremely dangerous, putting scientists at risk of discovering fashionable opinions rather than eternal truths, eroding all grounds for public reliance on science.

To talk more deeply about IIT is to engage in imaginary speculation, such as how the maximally independent conceptual structure might truly be different from the second-most-maximal one. With no data as a foundation, we are left making claims about the imaginary that objective reality cannot arbitrate. For this reason I’m cutting my investigation of IIT short here, and the whole community should follow suit. We need to look past IIT and focus on measures that can actually be taken, and their models thus refined, for science to move forward.

Footnotes

[1] IIT’s measure is based on what they call cause-effect repertoire, the possible causes and effects of the measured group of neurons being in a certain state (expressed as the values of neighboring neurons). How could we measure this? Do we hope that the brain settles into exactly the same state often enough that we can average the neighboring neurons to determine the probabilities? Do we manipulate the brain by changing neuron values to force a certain region to be the same over and over again? How do we determine the instantaneous state of the system that IIT requires, when there is no frame rate to the brain? What do we do about the fact that synaptic connectivity changes over time? Even with the perfect brain measuring device, additional breakthroughs would be needed to generate this data.

[2] The brain is estimated to have 86 billion neurons, imagine marking every neuron with a 1 (indicating it is in a particular power set) or a 0. How many different ways could you mark the neurons? As a 16-bit memory word has 2¹⁶ possibilities, these neurons could be marked 2⁸⁶ billion ways. 2⁸⁶⁰⁰⁰⁰⁰⁰⁰⁰⁰ ~= 10²⁵⁸⁰⁰⁰⁰⁰⁰⁰⁰.

[3] This complexity may seem whopping, but it gets even worse when we go from IIT’s simplified model of the brain (as a bunch of simple digital logic gates) to neurons themselves. Neurons are neither on nor off, but have firing rates that are more like continuous values. And of course synaptic connections are of various kinds. Instead of a binary 1 or 0, we would need a vector of floating point numbers to represent the state of a neuron. For this reason it’s impossible to even consider applying IIT to the brain without further extensions. While the authors imagine a future expansion of IIT to address these challenges, the opposite is what should happen. Instead of getting more complex, the theories that follow IIT need to be simpler so that we can actually perform measurements to verify them.

[4] It is also curious to find myself levying a subjective critique upon a scientific theory. Typically decisions amongst equations are decided by data, but here there is none at hand. This perhaps why IIT lingers: the foundation is so unmoored, it leads the conversation astray. It is said that arguing with idiots brings you down to their level. IIT is far from idiocy, but in a similar sense we are brought down to their axiomatic, “self-evident”, quasi-religious level when we engage. It doesn’t bother me that quantum theory is “curious” because it’s correct.

[5] In particular: if there is no relationship at all between an input to my wire, and its internal state, then we say there is no cause information. If the output of the wire can be strongly predicted using the internal state, then there is a large amount of effect information. The combination of these two causal entanglement measures makes up IIT’s cause-effect information.

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Steven Schkolne

South African/American Caltech CS PhD, turned international artist, turned questioner of everything we assume to be true about technology. Also 7 feet tall.