What is Mixed Reality? In short, Mixed Reality is the world which we now inhabit where real and virtual objects co-exist and can causally influence each other. In Mixed Reality persistent virtual objects can be created, seen, and manipulated and they exist and are perceived as integrated with the real world.

Together with the Internet of Things where environmental objects embed computational elements, sensors, and connectivity, Mixed Reality will radically alter the way we interact with and think about the world.

Reality as we know it just ended.

Origin Stories

In 1994 Paul Milgram and Fumio Kishino defined a Mixed Reality as “…anywhere between the extrema of the virtuality continuum.” (VC), where the Virtuality Continuum extends from the completely real through to the completely virtual environment with augmented reality and augmented virtuality ranging between.

“The conventionally held view of a Virtual Reality (VR) environment is one in which the participant-observer is totally immersed in, and able to interact with, a completely synthetic world. Such a world may mimic the properties of some real-world environments, either existing or fictional; however, it can also exceed the bounds of physical reality by creating a world in which the physical laws ordinarily governing space, time, mechanics, material properties, etc. no longer hold. What may be overlooked in this view, however, is that the VR label is also frequently used in association with a variety of other environments, to which total immersion and complete synthesis do not necessarily pertain, but which fall somewhere along a virtuality continuum. In this paper we focus on a particular subclass of VR related technologies that involve the merging of real and virtual worlds, which we refer to generically as Mixed Reality (MR).”

In 1993, I was leading the design of a concept for a Mixed Reality training system for the U.S. Army. The concept was to create a full body wireless interface to allow an individual combatant experience the virtual battlefield as if they were really there. Today you might implement something like this at home using Kinect and a Wii controller. But the DIVE Chamber concept was originally well beyond the state of the art.

Central to the project idea was that at the time there was a divided culture in the virtual reality community between those influenced by researcher Myron Krueger who emphasized the unencumbered and natural interaction of the body with virtual objects and the fully immersive head mounted display based approach pursued by pretty much everyone else. Another voice in the wilderness at that time was Brenda Laurel, whose book Computers as Theatre and concept of No Fucking Interface (NFI) suggested alternative ideas.

In her book, Laurel suggests the use of theatrical techniques and narrative to enhance virtual experiences. Her idea of NFI stated that visible or perceptible interface elements should be eliminated and interaction should be as natural as possible with control and interaction using the body, voice, and gaze as humans do in the real world. Both of these ideas shared the common idea of re-engaging the human themselves with the simulated experience in ways that went beyond mere visual surface appearances popularized in early VR demos.

The DIVE project proposed to build a specially designed room in which a user or participant could experience virtual reality. The virtual world was to be viewed through an HMD, actually an augmented reality display, and the room was to be equipped with video cameras for body tracking. The user would wear an HMD and experience audio via headphones with HRTF based 3D audio capability, and could also interact with the world through instrumented “props” such as weapons. Props were both physical objects that were held and virtual objects which could interact and cause effects within the simulated world. Originally we planned to use video based body tracking and a specially designed illuminated suit that could easily be tracked. We later abandoned this idea for off the shelf wireless magnetic tracking and replaced the headphone based audio with a high powered ambisonic speaker based system that could more realistically reproduce battlefield sounds such as gunfire and explosions at realistic decibel levels.

Integrating the props turned out to be much more complex than originally anticipated, for example, at this time there was no WiFi available so we had to hack a workable wireless solution together.

In order to wirelessly link the props to the virtual world we used an industrial wireless RS-232 data link developed for process control applications. Video was transmitted by a separate device. In order to appear realistic, the props had to be tracked along with the user’s head and body but also had to feature realistic controls such as trigger and safety mechanisms. This turned out to be problematic even using the then state of the art Datacube video rate image processing hardware we employed.

Batteries were also an issue, however, since our application was military training where soldiers normally carry gear and equipment we had an ample weight budget for extras like this. The illuminated suit itself required a battery of course and was dangerous. Bright colored LEDs did not exist at this time. We used fiber optics illuminated with a halogen lamp, and the lamp got hot requiring a custom box to cool it. Initial manned tests were attended with a fire extinguisher.

Props such as an M16 rifle, pistol, and a simulated Javelin anti-tank weapon were developed. Props were connected to a belt worn computer which processed raw signals and sent data back to the simulation computer via the wireless RS-232 link allowing for implementation of complex controls required for something like the Javelin.

We were focused around simulating firearms, but since the goal was a fully realistic immersive simulation of operations in urban terrain, we also had to consider hand grenades which are thrown. Some of these concepts have more recently been realized in a variety of systems, see below, where U.S. Marines are training with the Future Immersive Training Environment (FITE) and handgun training simulators using an omnidirectional mobility simulator known as a “hamster ball”.

Not Glass, A Great Leap Forward, and Microsoft Gets Their Mojo Back

The technology to support Mixed Reality is advancing rapidly and especially in the area of display technology. Two important new announcements were made recently in this area, one related to secretive start up Magic Leap and another, to the surprise of many, by industry stalwart Microsoft.

A full mixed reality experience requires use of a stereoscopic binocular head mounted display. This distinguishes these displays from the recently cancelled Google Glass device which was monocular and could not create immersive mixed reality experiences. This isn’t about being a Glasshole. It’s about changing the very nature of reality itself.

In early 2013 Canon made an augmented reality display commercially available that promised a full mixed reality experience. The Canon device did feature a stereoscopic display but required the use of special markers to aid the registration of real and virtual objects. This has been the state of the art in augmented reality technology for some time. Until now.

WIRED Magazine broke the story of mysterious startup Magic Leap’s patent application which included amazing concept art of a Mixed Reality world populated by magical interactive objects integrated with the normal physical world.

The Magic Leap display is a see through stereo-augmented reality display with a belt worn controller similar to that envisioned in the DIVE project. The Magic Leap concept suggests a deep Mixed Reality world where interfaces and displays materialize in mid air or around the body as desired and in which users can interact with photorealistic virtual objects and characters.

Body centric interfaces allow controls and displays to appear anywhere as needed, and allow the user to interact directly with their entire body and without the need for specialized clothing or suits beyond the display and a glove shown for gesture control. Virtual objects can be registered with real objects in the world like tables or “slates” made out of simple flat pieces of metal or other materials. An appropriately mysterious object in the form of a mixed reality egg was shown.

The tech world was already buzzing with the dreams of Magic Leap’s amazingly envisioned mixed reality world, when Microsoft dropped the bomb of their HoloLens project. Microsoft is officially back and it’s big.

HoloLens is seemingly already delivering the promise of a true mixed reality experience today. Both WIRED and C-NET report hands on experience with the device describing it as jaw dropping and amazing.

Microsoft’s product goes beyond the obvious. Sure you can interact with Windows via menus and displays that appear to float in front of you. But also, you can play Minecraft integrated into the real world. Suddenly, Microsoft’s somewhat surprising move on Mojang, creators of the game, makes all the sense in the world.

In addition to the Minecraft world, Microsoft naturally showed some impressive practical and business oriented applications including virtually helping someone repair a sink, use of mixed reality in design, and more. HoloLens isn’t just for games, it’s an entirely new way to interact with machines.

Mixed Reality is Really a New World

The virtual world is now real, and the real world is now virtual. Our new Mixed Reality environment combines technologies from multiple areas and integrates various infrastructures into a unified experience. It includes and integrates augmented reality displays, wearable technologies and the quantified self, the Internet of Things, the Internet of Me, and well, pretty much anything else you can think of.

But Mixed Reality isn’t just a Silicon Valley gimmick, it’s an entirely new and unique mode of existence where real and virtual merge.

In a 2011 paper Alfred Hubler and Vadas Gintautas experimentally demonstrate “synchronized states”, where “the boundary between the real system and the virtual system is blurred”. Their apparatus uses physically coupled virtual and real systems , in this case real and virtual pendulums, but similar arrangements could be envisioned between more complex dynamical systems. Hubler and Gintautas show that the system of coupled pendulums exists in a mixed reality state which is not entirely real and not entirely virtual but both and in between.

In order to create this effect, an instantaneous, or nearly so, bidirectional coupling is required between real and virtual objects. When this coupling occurs the result is a mixed reality state. Mixed Reality entails, therefore, causal couplings between real and virtual objects. Eventually we will be surrounded by many such causally coupled systems and objects.

An important property of mixed reality systems is that their behavior can not be determined by examination of either the virtual or physical world alone. Since the state of the system is coupled both to physical and virtual realities, examining either one alone can not capture these “hidden” causal linkages.

For example, a home management and control system presented in a Mixed Reality environment would connect virtual objects used to control features like temperature and lighting, as well as sensors in the actual physical environment. The result is that your future augmented reality controlled home will probably exist in a mixed reality state. You might not even notice it though.

Dangers: Reality Distortions and Disorders

No technology is entirely risk free. Mixed Reality seems amazing and holds promise to radically extend human abilities to understand and act in the world. But what happens when it gets too good? When virtual objects can not easily be distinguished from real ones?

And what happens when you take the device off and return to the information poor and non-magical ordinary reality we inhabit today? Will we ever want to take these off?

Certainly we can imagine some people that already have difficulties distinguishing reality from fantasy whom may also have issues with Mixed Reality environments. But what happens to all of us when reality is fundamentally influenced by the virtual world? Seemingly effectively operating in such a full Mixed reality scenario requires that you remain connected to the shared Mixed Reality environment at all times. Eventually, Mixed Reality may become simply known as “reality”.

Another danger, Mixed Reality might be annoying. Advertisers can now know not only where you are but what you are looking at. Virtual advertisements can appear in your real world and in your private spaces, “pushed” to your display. The idea was pioneered in the development of NanoHome a long lost 3D world which I led the development of before co-founding Live365.com. In the NanoHome world everyone had their own 3D virtual mansion which they could decorate and use to share media content. But we could insert objects, for example a dancing soda can could appear on your kitchen table and the environment knew where you looked and for how long. With Mixed Reality, add eye tracking and gaze direction to the long list of things the government can monitor about you. And of course the device also constructs and stores a 3D model of your immediate surroundings.

Mixed Reality is the Interface to Magic

But the possibilities out weigh these negatives perhaps.

Once you understand the full implications of Mixed Reality as a separate “interreality” physical state, you inevitably also realize that it implements many of the ancient ideas of magical manipulation of reality.

Using just their voice, various coded gestures and movements, along with computational symbolic systems, the user is able to create and manipulate objects in both an imaginary world of the mind and the physical material realm. As the world becomes populated with interactive “thinking” systems, these will be controllable from Mixed Reality. They will be mixed reality objects which exist in both real and virtual realms at once. Virtual controllers will influence and cause real physical effects sometimes in surprising ways. Mixed Reality will become the interface through which we interact with the Internet of Things, talk to robots, drones, smart buildings, intelligent vehicles and eventually nanotechnology such as nanorobotic swarms. And that’s just the start of what is seemingly possible.

Mixed Reality is the interface to magic.

It isn’t just because imaginary objects and avatars can now appear and penetrate our realities. Beyond Mixed Reality lies a hidden realm. The notion was explored in another early virtual reality project we did for the U.S. Army called Object Oriented VR. Every object in a virtual world consists of both a visual representation, what it appears to be, and a data representation which includes the various data elements that control the object’s behavior in the simulated world. However, these data items can be presented as physical or visual objects themselves in the Mixed Reality world. Even the controlling software itself can become visually and materially present in Mixed Reality. And you can toggle between realms with the raising of a finger, and they can be overlaid, layered, and simultaneously present as well.

The notion isn’t very different from what some users of DMT report experiencing in their interactions with the so called “machine elves” that some say construct our world.

But in the new world of Mixed Reality it’s true.

Beyond the mundane world lies a magical realm, the realm of code, the realm of pure mathematics and computation, also the world of imagination, the world where anything can become real and where information manages and controls the levers of our new Mixed Reality environment.

The idea was presented most effectively perhaps in Char Davies’ transformative VR experience Osmose. In Osmose, the underworld of code is the secret world that lies beneath the roots of a tree surrounded by and penetrated by floating orbs of light. The user experiences the world through an immersive head tracked head mounted display along with algorithmically generated music.

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Finally, it is important to remember that the virtual world need not obey the existing laws of physics of the mundane one. Virtual objects are limitless, do not need to obey the laws of gravity or conservation of momentum. Virtual objects can hold entire worlds, and a space can contain an even larger space within it. Objects can hide controls and interfaces whether they are real or virtual. A flower can become a controller, with catching falling petals giving you one last chance to restore your deleted files. Imagination is the limit.

Reality just became “mixed” , a little bit more strange, and possibly a whole lot more interesting.

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Peter Rothman, in addition to being editor of h+ Magazine, holds a B.A. in mathematics from the University of California at Santa Cruz and a M.S. in computer engineering from the Virterbi School at the University of Southern California. He studied chaos theory, dynamic and nonlinear systems theory, general systems theory, Artificial Intelligence, and neural networks. He designed a dual mode (analog/digital) neural processing system for DARPA especially developed for recurrent multilayer networks such as those employed in deep learning architectures.

Originally published at hplusmagazine.com on January 22, 2015.

Is infinity an illusion?

Strict finitism, or simply finitism, is a somewhat obscure mathematical idea that has important implications for transhumanism and key transhumanist projects such as mind uploading and artificial general intelligence. Finitism is a philosophy of mathematics that rejects the existence of infinite mathematical objects.

The finitist philosophy of mathematics is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects like infinite sets are accepted as legitimate objects existing in some sort of “Platonic universe” of forms.

For example, strict finitists accept finite sets of natural numbers, rational numbers and finite precision real numbers, but they do not accept infinite sets and infinite series, real numbers, surreal numbers, transfinite numbers, etc. Importantly a strict finitist would reject the idea of a Turing Machine in the general case because it includes the idea of an infinitely long tape. For a finitist ideas like Turing Machines, advice TMs, analog neural networks, and similar systems are essentially nonsensical or, at best, non-realizable ideals or thought experiments.

Within mathematics, the finitists are contrasted with the transfinitists who are proponents of Georg Cantor‘s hierarchy of infinities. Leopold Kronecker was perhaps the best known finitist and opponent to Cantor’s set theory who said,

God created the natural numbers, all else is the work of man.

Reuben Goodstein is another well known proponent of finitism and his work involved building up to analysis from finitist foundations. Although he apparently denied it, much of Ludwig Wittgenstein‘s writing on mathematics also has a strong affinity with finitism.

Aristotle might be characterized as a strict finitist. However Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity:

But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. If, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in.

— Aristotle, Physics, Book 3, Chapter 6

Aristotle’s actual infinity means simply an actualization of something never-ending in nature, which is in contrast the Cantorist actual infinity which means that of the transfinite cardinal and ordinal numbers, which has nothing to do with things in nature.

Ultrafinitism (also known as ultraintuitionism) has an even more conservative attitude towards mathematical objects than finitism, and has objections to the existence of finite mathematical objects when they are too large to be produced or considered.

Physical finitism is the position that all observable things and entities are finite systems. Notably since humans are finite systems in both time and space, infinite quantities, sets, and objects can not be taken to meaningfully reference humans or to exist since humans being are finite beings that can never produce or examine infinite objects.

The simplest argument against finitism is simply the existence of the elementary mathematical operations. For example addition of one produces the next number in sequence, and so it seems, as long as we keep adding, we always get a number, ad infinitum.

However physical finitism suggests that according to any method we might choose, and however diligently and long we work, we’ll never get an infinite quantity. The result is always a finite number, one more than the last. Eventually we would get to a number so large we couldn’t record it using all of the matter in the universe. It’s not entirely clear what it means for a number larger than this to exist.

Even if one rejects temporal finitism, the idea that time is finite, we observe that individual organisms and intelligent systems have finite life spans. Knowledge is finite, as is the set of all things we can imagine knowing. What we know and don’t know is limited by our use of language, which is by the way, always a finite system. THis relationship might even point to a deeper relationship between these ideas of time, space and knowledge.

How Smart Are Humans Really?

One of the obvious but controversial conclusions of physical finitism is that humans are not even Turing Machines, but in fact, seemingly, just very complex finite state machines. Like your thermostat or TV remote control, but more so.

Consider, the definition of a Turing Machine allows for an infinite tape with finite control logic. But humans, since they are finite systems, can not store an infinite amount of information. We’ve got a finite tape.

We only live for a finite amount of time, and can only process a limited amount of sensory input per unit time. In other words, the number of things we can experience is limited by how long we live and how many experiences we can have per second. Each experience produces a brain state and pattern of memories, but only finitely many of these can exist.

If you think of yourself as a Turing Machine, you can only be a finite Turing Machine. Too bad, because this limits the sorts of computations that humans can do unaided and by a lot. But even a true Turing Machine running a finite amount of instructions per second can only process a finite amount of memory in a finite time.

So, according to finitism, humans can not be Turning Machines but must in fact be merely complex automatons.

The idea has some pretty significant implications for the limitations of human knowledge, creativity, and inventiveness that humans may find a bit hard to swallow in the future. If humans are nothing more than large finite state machines then any output or creative activity we can produce must, by definition, be constrained by a regular language.

We are not as smart as we imagine ourselves to be.

What About the Singularity and Artificial Intelligence?

All existent artificial intelligence systems are finite computer programs running on finite machines, and therefore, are not truly Turing Machines at all but also finite state machines or “finite memory” Turing Machines. Despite what you might have (mistakenly) learned in computer science class, all real world computers are only finite Turing Machines too, just like humans.

However, a well known fact, any finite Turing Machine can be shown to be equivalent to a (possibly very large) deterministic finite state machine.

Importantly, the well known halting problem is decidable for finite Turing machines and all FSMs but possibly only in theory as deciding may take as long as executing the number of states considered.

Here is a non-formal outline of the proof:

• A Turing machine with a finite amount of memory can only be in a finite number of configurations. A configuration consists of the position of the head on the tape and the state of the machine. Since both the number of states and the number of locations on the tape are finite, this is also a finite number which we will call N.
• After running N+1 instructions, either the machine must stop, or you must visit the same configuration, by the so called “pigeon-hole” principle, creating a loop. Thus, after running N+1 instructions if the machine has not yet halted, you conclude that it won’t ever halt, but instead loops forever.

Since any realizable machine must obey the Bekenstein Bound, it must process a finite amount of information and can only consume a finite amount of energy. A realizable machine can only contain a finite amount of matter and therefore can only store finite programs and inputs.

Theorem (Bekenstein 1981): A spherical region with radius R and energy E can contain only a limited amount of information I (in the sense of number of distinguishable (quantum) states):

I ≤ 2π ER/¯hc ln 2 where ¯h is Planck’s constant and c is the speed of light.

Alternatively, I ≤ kMR where M is the mass in the region and k ≈ 2.57686 · 1043 bits/(m kg).

No Foom

Importantly strong finitism rules out the existence of an infinite intelligence explosion in our universe. All physical systems are limited in power to that of finite automata. Like any quality of a finite system, finitists assert that intelligence must also remain strictly finite.

Given any self improving intelligent system that has existed for time t1, there is some finite amount of space/time that is reachable by the system at time t2. Thus our system can only consider a finite number of sensory inputs or experiences, can only store a finite number of memories, etc. and the number of possible configurations must therefore also remain finite.

No one really knows exactly how much information is required to accurately represent a human brain, however, finitism suggests that it must of course be a finite quantity.

So, given that we observe rapid and exponential or super-exponential advances in information processing and information storage technology, finitism supports the idea of a technological Singularity in the form of the emergence of a greater than human machine intelligence.

In fact, a significantly “larger” intelligence can produce objects which appear infinite to us, and this has important consequences. A much greater intelligence can for example algorithmically construct sequences which are indistinguishable from random bits to humans since the program required to generate them can not be executed by any human brain.

Intelligence augmentation of course would radically alter the size of programs human can “run”.

But without it, a much greater than human superintelligence could produce mathematical proofs that humans can not evaluate manually for correctness as another example. This is already happening with existing work in computer based proofs where proof generating programs either have to be derived from provably correct principles or, at least, the results are checked by a different program or process later.

So while finitism does rule out a formally infinite intelligence explosion, it doesn’t in any way rule out a relative and apparently infinite intelligence explosion. A much greater than human intelligence could, for example, seem to produce an infinite number of digits for any computable irrational number we select, e.g. it would appear to us to know all of the digits of pi despite knowing only a finite number of them. The chance of us guessing one it didn’t know, would be vanishingly small.

According to physical finitism, all infinite processes are strictly ruled out. A machine, no matter how intelligent, can only reach a finite region of spacetime during its lifetime and therefore can only consume a finite amount of energy and process only a finite amount of information.

All of the universe which is observable to us, lies inside the region of spacetime known as our light-cone.

Consider an intelligence called “A” located at a point in space-time labeled here ‘A’.

In the plot below, time is graphed on the vertical axis, and space is in the imagined horizontal plane. The speed of light defines two cones, one into A’s past and one into A’s future. However, note that various places lie outside of IAs past and future light-cones. Events outside of A’s light-cone, e.g. the event labeled E, can not have any causal influence over A’s structure, state or dynamics.

All relevant information to A’s state must be contained in A’s past light-cone.

Infinite minds, according to finitism, do not exist, but vast planet sized “Jupiter Brains” still might.

Hyperwhat?

Some researchers claim the mind is a type of super Turing or hypercomputation, that is, a machine beyond any Turing Machine.

Note that we can make only a finite number of observations of limited precision of any physical system. This implies that the claim that some system X , say me, is a Super Turing Machine or even simply a Turing Machine may not be empirically meaningful or decidable since there is no method to determine this either way in finite time. It’s an unfalsifiable assertion.

Given that the notion of universal computation is defined over an infinite space of functions, it would appear that no finite series of measurements could ever reject the hypothesis that a real physical system was only computationally equivalent to a finite automaton as opposed to some variety of universal computer. In other words, the hypothesis that a real physical system such as a human cognizer has universal computational power does not appear to be empirically meaningful.

~ Gregory Mulhauser

(see also Russell’s Teapot)

Implications for Mind Uploading

If the human mind is a finite information processing system of some sort, and it certainly seems that it is, then it can process at most a finite amount of information, it can store only a finite number of memories, and it can only perform a finite amount of computation during its lifetime. A full life image or complete sensory life log would include at most a finite number of bits. A full “mindclone” would therefore also require at most a finite number of bits.

Simply stated finitism supports and suggests the viability of mind uploading since a mind, whatever it might be, must exist in a finite region of spacetime and must be defined by a finite amount of information.

In fact it seems that the human mind is a rather simple sort of machine, simpler than the classical Turing Machine from CS 101.The mind it seems is a discrete state finite automation sometimes also known in the literature as an FSM or DFA.

This view is controversial, and various alternative structures have been introduced to counter opposing arguments. Notably by David Chalmers who extends the notion of a finite state automaton to what he terms a combinatorial state automaton. Others have tried similar maneuvers, however, often these alternative formulations can be shown to be equivalent to either FSMs or TMs.

Nevertheless, the idea that the mind is a finite state machine, however vast, has obvious and important implications for mind uploading. In theoretical computer science a bisimulation is a binary relation between state transition systems, which associates systems that behave in the same way. Intuitively two systems are bisimilar if they match each other’s moves during dynamic operation of both systems. In this sense, the systems cannot be distinguished from each other by an outside observer.

In some cases finite bisimulations can be shown to exist for hybrid systems, allowing even complex continuous behaviors to be analyzed as a bisimulation of a finite system. Bisimulation therefore provides a rich mathematical framework on which to analyze proposals for uploading, mindcloning and whole brain emulation. For example, there exist some results about the computational complexity of bisimulation processes. Discussing these is beyond the scope of this article however.

If we can build a bisimulation of your mind, it would be indistinguishable from you. It would think the same thoughts, dream the same dreams, make the same mistakes.

Wait a Minute There

Isn’t the universe infinite?

Well, no one really knows whether the universe is unbounded and infinite or unbounded and finite (closed). At the very least the universe is extremely large and most of the universe remains outside of your light-cone and that of any imaginable intelligence that might originate on Earth. So even a superintelligence, where it to start here, has to remain “small”.

Another seeming problem with finitism is that it possibly violates quantum mechanics such as the no cloning theorem. It seems at least first glance that it is entirely possible to copy a finite system. DOes this imply that the mind is not a quantum system? Finitist proponents would object to any non-finite description of mind whatever mind might really be. This will invigorate proponents of whole brain emulation, but aggravate advocates of quantum mind theories. Finitists think that the human mind must be a sort of regular language.

Some physicists theorize that the universe is a discrete lattice and that continuous space does not exist and some physical experiments support this idea. And of course the speed of light is exactly the sort of physical limitation that a finitist viewpoint would suggest.

Finitism at least suggests that we question whether we can know anything about claimed infinite quantities or qualities. We imagine π has an infinite number of digits, however, we can never produce evidence that this is the case. An infinite decimal expansion can not be recorded in the accesible universe. While we can produce a method that seems to produce an infinite series, there is no executable process or procedure to construct an infinite object or test members of an infinite set for a characteristic or quality.

All real things have at most a finite amount of time and energy available to them.

At best, we can approximate infinite ideas with a finite quantity or measure. But, also importantly, in some cases we can produce the answer to a question that seems to involve an infinite procedure in a finite number of steps. For example, in the mathematics of limits a seemingly infinite limit process is replaced with a finite series of steps by which the human mathematician arrives at the correct answer and without the need to do infinitely many steps.

Conclusions

Even if strong finitism is incorrect and some sort of infinite space or structure does indeed “really” exist, caution is advised when treating what are just very complex finite systems as infinite ones. We can, for example, copy finite sequences but not infinite ones. We might mistakenly conclude something like mind uploading is impossible as a result.

While we can’t implement an infinite procedure, in some cases, we can convert infinite procedures into finite ones. Seemingly infinite quantities and qualities can be represented by finite symbols and manipulated by finite brains. So we can think about infinity with finite resources, in finite time, and even with our seemingly not that amazing monkey minds. Perhaps there is hope for us yet.

If the mind is indeed just a complex finite system, this improves the viability of many transhumanist projects. It suggests bisimulation as a mathematical framework for constructing, testing, and evaluating uploaded minds and mindclones. It outlines some realistic bounds on computation required to construct whole brain emulations.

Whatever position you take on finitism, this obscure area of the philosophy of mathematics is well worth understanding.

References

van der Velde, Frank. “Is the brain an effective Turing machine or a finite-state machine?.” Psychological research 55, no. 1 (1993): 71–79.

Levelt, Wilemm JM. “Are multilayer feedforward networks effectively Turing Machines?.” Psychological Research 52, no. 2–3 (1990): 153–157.

Lokhorst, G. J. “Why I am not a super-Turing machine.” In Hypercomputation Workshop, University College, London, vol. 24. 2000.

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Peter Rothman, in addition to being editor of h+ Magazine, holds a B.A. in mathematics from the University of California at Santa Cruz and a M.S. in computer engineering from the Virterbi School at the University of Southern California. He studied chaos theory, dynamic and nonlinear systems theory, general systems theory, Artificial Intelligence, and neural networks. He designed a dual mode (analog/digital) neural processing system for DARPA especially developed for recurrent multilayer networks such as those employed in deep learning architectures.

Originally published at hplusmagazine.com on December 16, 2014.