## Introduction

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 human beings 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.

~ 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.