Very large numbers are not numbers: Infinity does not exist

(this blog post was originally published in . It has been moved to this blog – with slight changes.)

Remember Nietzsche’s famous announcement, “God is dead“? In the domain of mathematics, Nietzsche’s announcement could just as well refer to infinity.

There are some philosophers who are putting up a major challenge to the Platonic stronghold on math: Brian Rotman, author of Ad Infinitum, is one of them. I am currently reading his book. I thought of waiting until I was finished with the book before writing this blog post, but I decided to go ahead and splurt out my thoughts.


Charles Petzold gives a good review of Rotman’s book here.

Petzold says:

“We begin counting 1, 2, 3, and we can go on as long as we want.

That’s not true, of course. “We” simply cannot continue counting “as long as we want” because “We” (meaning “I” the author and “you” the reader) will someday die — probably in the middle of reciting a very long (but undoubtedly finite) number.

What the sentence really means is that some abstract ideal “somebody” can continue counting, but that’s not true either: Counting is a temporal process, and at some point everybody will be gone in a heat-dead universe. There will be no one left to count. Even long before that time, counting will be limited by the resources of the universe, which contains only a finite number of elementary particles and a finite amount of energy to increment from one integer to the next.”

Is Math a Human Activity or Eternal Truth?

Before continuing on to infinity (which is impossible of course), I want bring up a related topic that Rotman addresses: the nature of math itself. My thoughts at the moment are this:

You (reader) and I (writer) have brains that are almost identical as far as objects in the universe. We share common genes, language, and we are vehicles that carry human culture. We cannot think without language.  “Language speaks man” – Heidegger.

Since we have not encountered any aliens, it is not possible for us to have an alien’s brain planted into our skulls so that we can experience what “logic”, “reality” or “mathematical truth” feels like to that alien (yes, I used the word, “feel”). Indeed, that alien brain might harbor the same concept as our brains do that 2+2=4….but it might not. In fact, who is to say that the notion of “adding” means anything to the alien? Or the concepts of “equality”? And who is to say that the alien uses language by putting symbols together into a one-dimensional string?

More to the point: would that alien brain have the same concept of infinity as our brains?

It is quite possible that we can never know the answers to these questions because we cannot leave our brains, we can not escape the structure of our langage, which defines our process of thinking. We cannot see “our” math from outside the box. That is why we cannot believe in any other math.

So, to answer the question: “Is math a human activity or eternal truth?” – I don’t know. Neither do you. No one can know the answer, unless or until we encounter a non-human intelligence that either speaks an identical mathematical truth – or doesn’t.

Big Numbers are Patterns

My book, Divisor Drips and Square Root Waves, explores the notion of really large numbers as characterized by pattern rather than size (the size of the number referring to where it sits in the countable ordering of other numbers on the 1D number line). In this book, I explore the patterns of the neighborhoods of large numbers in terms of their divisors.

This is a decidedly visual/spatial attitude of number, whereby number-theoretical ideas emerge from the contemplation of the spatial patterning.

The number:


doesn’t seem to have much meaning. But when you consider that it is the number of ways in which you can arrange a single deck of cards, it suddenly has a short expression. In fact it can be expressed simply as 52 factorial, or “52!”.

So, by expressing this number with only three symbols: “5”, “2”, and “!”, we have a way to think about this really big-ass number in an elegant, meaningful way.

We are still a LONG way from infinity.

Now, one argument in favor of infinity goes like this: you can always add 1 to any number. So, you could add 1 to 52! making it 80658175170943878571660636856403766975289505440883277824000000000001.

Indeed, you can add 1 to the estimated number of atoms in the universe to generate the number 1080 + 1. But the countability of that number is still in question. Sure you can always add 1 to a number, but can you add enough 1’s to 1080 to each 10800?

Are we getting closer to infinity? No my dear. Long way to go.

Long way to “go”?  What does “go” mean?

Bigger numbers require more exponents (or whatever notational schemes are used to express bigness with few symbols – Rotman refers to hyper-exponents, and hyper-hyper-exponents, and further symbolic manipulations that become increasingly hard to think about or use).

These contraptions are looking less and less like everyday numbers. In building such contraptions in hopes to approach some vantage point to sniff infinity, one finds a dissipative effect – the landscape becomes ever more choppy.

No surprise: infinity is not a number.

Infinity is an idea. Really really big numbers – beyond Rotman’s “realizable” limit – are not countable or cognizable. The bigger the number, the less number-like it is. There’s no absolute cut-off point. There is just a gradual dissipation of realizability, countability, and utility.

Where Mathematics Comes From

Rotman suggests taking God out out mathematics and putting the body back in. The body (and the brain and mind that emerged from it) constitute the origins of math. While math requires abstractions, there can be no abstraction without some concrete embodiment that provides the origin of that abstraction. Math did not come from “out there”.

That is the challenge that some thinkers, such as Rotman, are proposing. People trained in mathematics, and especially people who do a lot of math, are guaranteed to have a hard time with this. Platonic truth is built in to their belief structure. The more math they do, the more they believe that mathematical truth is discovered, not generated.

I am sympathetic to this mindset. The more relationships that I find in mathematics, the harder it is to believe that I am just making it up. And for that reason, I personally have a softer version of this belief: Math did not emerge from human brains only. Human brains evolved in Earth’s biosphere – which is already an information-dense ecosystem, where the concept of number – and some fundamental primitive math concepts – had already emerged. This is explained in my article:

The Evolution of Mathematics on Planet Earth

I have some sympathy with Roger Penrose: when I explore the Mandelbrot Set, I have to ask myself, “who the hell made this thing!” Certainly no mathematician!

After all, the Mandelbrot Set has an infinite amount of fractal detail.

But then again, no human (or alien) will ever experience this infinity.


How much negentropy is Earth capable of?

Negentropy is the opposite of entropy. It refers to an increase in order, complexity, and usefulness, while entropy refers to the decay of order or the tendency for a system to become random and useless.

The universe as a whole tends toward total entropy, or heat death. This does not mean that ALL parts of the universe are becoming less ordered. There can be isolated parts of the universe that are actually increasing in order; becoming more organized and workable. The best example of this is our home: planet Earth.

A miracle of 7,000,000,000,000,000,000,000,000,000 atoms

I was walking from my bedroom to my bathroom this morning, pondering the miracle of my body purposefully moving itself from one place in the universe to another. Consider the atoms that make up my body; they are assembled in just the right way to construct a human capable of locomotion. It is a miracle. Of course, the atoms themselves are not the driving force of this capability. The driving force is a collaboration of emergent systems (molecules, tissues, electrochemical activity, signals between organs, and of course, a brain – which evolved in the context of a complex planet, with other brains in societies, and with an ever-complexifying backdrop of shared information.

It’s a curious thing: planet Earth – with its vast oceans, atmosphere, ecosystems and organisms – is determined to go against the overall tendency in the universe to decay towards the inevitable doom of heat death.

While walking the seven billion billion billion atoms of my body to the bathroom, I considered how far the negentropic urge of our planet could possibly push itself, in a universe that generally tries to ruin the party; a universe that will ultimately win in the end. The seven billion billion billion atoms currently in my body will eventually be strewn throughout a dead universe. At that point there will be nothing that can re-assemble them into anything useful.

How not to ruin a party

The party is not over; there is ample reason to believe that Earth is not done yet. Earth generated a biosphere – the only spherical ecosystem we know of – which produced animals and humans, and most recently – post-biological systems (technology and AI). I would not dismiss entirely the notion that Earth really wants us to invent AI, and to allow it to take over – because our AI could ultimately help Earth stay healthy, and continue its negentropic party. We humans (in our old, biological manifestation) are not capable of taking care of our own planet. We are only capable of exploiting its resources – left to our own primitive survival devices. It is only through our post-human systems that we will be able to give Earth the leverage it needs to continue its negentropic quest.

This is another way of saying that the solutions to climate change and mass extinction will require massive social movements, corporate and governmental leadership, global-scale technologies, and other trans-human-scale systems that far exceed the mental capacities of a single human brain. It is possible that the ultimate victory of AI will be to save ourselves from an angry Mother on the verge of committing infanticide.

In the meanwhile, Earth may decide that it needs to get rid of the majority of the human population; just another reason to reconsider the urge to make babies.

But just how far can Earth’s negentropic party extend? As Earth’s most potent agents of negentropy, we humans are preparing to tap the moon, asteroids, and other planets for resources. Will we eventually be able to develop energy shields to deflect renegade asteroids? Will our robots continue to colonize the solar system? How far will Earth’s panspermia extend?

There are plenty of science fiction stories and hypothetical explorations that offer exciting and illuminating possible answers to these questions; I will not attempt to venture beyond my level of knowledge in this area. All I will say is…I think there are two possible futures for us humans:

(1) Earth will decide it has had enough of climate change, and smack us down with rising oceans and chaotic storms, causing disease, mass migrations, and war, resulting in our ultimate demise (Earth will be fine after a brief recovery period).

(2) We will evolve a new layer of the biosphere – built of technology and AI – and this will regulate our destructive instincts, thus allowing Earth to stay healthy and to keep complexifying. It will allow Earth to reconsider what it currently sees as a cancer on its skin – and to see us as agents of health.

In the case of future (2), we will lose some of our autonomy – but it just might be a comfortable existence in the long run – because Earth will be better off – and it will want to keep us around. Eventually, the panspermic negentropic party will not be our own – we will be just one of the intermediate layers of emergence emanating from the planet. We will become mere organs of an extended post-Earth ecosystem that continues to defy the general entropy of the universe…at least for a few billion more years.

The feeling of consciousness is an illusion

Stanislaw Dehaene’s book, Consciousness and the Brain, identifies various kinds of consciousness. It helps to separate the various uses of the words “conscious” and “consciousness”. The kind of consciousness that he has studied and reported in his book has measurable effects. This allows the scientific method to be applied.

After reading Dehaene’s book, I am more convinced that science will eventually fully explain how we hold thoughts in our minds, how we recognize things, form ideas, remember things, process our thoughts, and act on them. To be conscious “of” something – whether it be the presence of a person, a thing, or a fleeting thought – is a form of consciousness that can have a particular signature – physiological markers that demonstrate a telltale change in the brain that coincide with a person reporting on becoming aware of something.

Brain imaging will soon advance to such a degree that we will begin to see signatures of many kinds of thoughts and associate them with outward behaviors and expressions. It it also being used to show that some people who are in a vegetative state are actually aware of what is going on, even if they have no way to express this fact outwardly. So much will be explained. We are at a stage in brain research where consciousness is becoming recognized as a measurable physical phenomenon. It is making its way into the domain of experimental science. Does this mean that consciousness will soon no longer be a subject of philosophy?


There is one kind of consciousness which we may never be able to directly measure. And that is the subjective feeling of being alive, of being “me”, and experiencing a self. It is entirely private. Daniel Dennett suggests that these subjective feelings, which are referred to as “qualia”, are ineffable: they cannot be communicated, or apprehended by any other means than one’s own direct experience.

This would imply that the deepest and most personal form of consciousness is something that we will never be able to fully understand; it is forever inaccessible to objective observation.

On the other hand, the fact that I can write these words and that you can (hopefully) understand them means that we probably have similar sensations in terms of private consciousness. The vast literature on personal consciousness experience implies a shared experience. But of course it is shared: human brains are very similar to each other (my brain is more similar to your brain than it is to a galaxy, or a tree, or the brain of a chicken or the brain of a chimp). The aggregate of all reports of this inaccessible subjective state constitutes a kind of objective truth – indirect and fuzzy, sure – but nonetheless a source for scientific study.

So I’d like to offer a possible scenario that could unfold over the next several decades. What if brain scientists continue to map out more and more states of mind, gathering more accurate and precise signatures of conscious thoughts. As more scientific data and theories accumulate to explain the measurable effects of consciousness in the brain, we may begin to relegate the most private inexpressible aspects of qualia to an increasingly-smaller status. Neuroscience will enable more precise language to describe subtle private experiences that we have all experienced but may not have had a clear way to express. Science will nibble away at the edges.

An evolved illusion

And here’s an idea that I find hard to internalize, but am beginning to believe:

It’s all an illusion.

…because self is an illusion; a theatre concocted by the evolving brain to help animals become more effective at surviving in the world; to improve their ability to participate in biosemiosis. Throughout evolution, the boundary between an organism’s body and the rest of the world has complexified out of necessity as other organisms complexify themselves – this includes social structures and extended phenotypes. Also, the more autonomous the organisms of an evolving species become, the more self is needed to drive that autonomy.

The idea that we are living in an illusion is gaining ground, as explored in an article called: “The Evolutionary Argument Against Reality“.

Feelings are created by the body/brain as it interacts with the world, with thoughts generated in the brain, and with chemicals that ebb and flow in our bodies. The feeling of consciousness might be just that: a feeling – a sensation – like so many other sensations. Perhaps it was invented by the evolving brain to make it more of a personal matter. The problem is: being so personal is what makes it so difficult to relegate to the status of mere illusion.

The sleeping sponge: on the evolution of waking up

From the book, Wide Awake at 3:00am, I learned that researchers had come up with an answer to a common question, “Why do we sleep”?

It’s a valid question. What’s the actual purpose of sleep? Why would nature favor having the majority of animal species waste several hours each day in a state of unconsciousness, getting nothing done, and becoming vulnerable to predators?

The answer the researchers came up with required turning the question on its head: “Why should any living thing bother waking up at all?” Perhaps sleep is the normal state of all life, and wakefulness is just some aberration – a phenomenon that evolved later – as a part-time activity to more efficiently pursue food and sex.

As a lover of naps and hater of alarm clocks, I kind of like this idea.

I recall reading somewhere that sponges are “always asleep”. But I also read recently that sponges “never sleep”. Rather than go back and do more research to clear up this issue, I shall instead declare that the problem lies the definition of  “sleep”.

If you’re a sponge, you have no neurons. Having no neurons is a good indication that you have no brain. And no brain means no dreaming. Sponges are not like us in that they are sessile: they have no motility (except in the larval stage, when genetic dispersal occurs). If you don’t have to get up and go to work, why bother having a brain? Brains provide inner-representations of the outside world – used to navigate unpredictable terrains. Sponges just sit there at the bottom of the ocean and collect ambient nutrition. A task so easy that anyone can do it in their sleep.

Brains for Movement

The evolution of mobility required not only the direct control of muscles but also representations of reality that determined when and how those muscles get activated. Brains evolved in order for animals to evolve.

Long ago, there was no such thing as “waking up”. Until brains came along and gave organisms a reason to get off their asses and get a job. Perhaps asses and jobs had to evolve as well. But let’s not get too technical here.

It is possible that the binary states of wakefulness and sleep were not invented by brains themselves, but earlier in evolutionary history, by simple neuronal networks that generate sleep-like dynamics. Given that every location on Earth other than the poles has been cycling between day and night since before life emerged, it makes sense that organic periods would emerge to harmonize with this cycle.

Perhaps the very process of storing representations of reality – no matter how small or simple – requires a periodic cycle – as indicated by research finding that sleep is required for brains to prune useless memories and absorb useful ones.

My takeaway from all of this is that I have an organ that likes to make me do complicated things for many hours each day: sixteen to be exact. That’s a long time each day being on the move and getting worked up about other brains that are wreaking havoc on the world, such as the shriveled-up shitball inside of Donald Trump’s skull.

Before I die, I will thank my brain for collecting a massive library of memories that fueled a lifetime of dreams. And then I will say goodnight to my brain, and get back to sleep.

Here’s one way to evolve an artificial intelligence

This picture illustrates an idea for how to evolve an AI system. It is derived from the sensor-brain-actuator-world model.

Machine learning algorithms have been doing some impressive things. Simply by crawling through massive oceans of data and finding correlations, some AI systems are able to make unexpected predictions and reveal insights.

Neural nets and evolutionary algorithms constitute a natural pairing of technologies for designing AI systems. But evolutionary algorithms require selection criteria that can be difficult to design. One solution is to use millions of human observers as a Darwinian fitness force to guide an AI towards an optimal state.


Since there is so much discussion (and confusion) on AI these days I want make some clarifications.

  • This has nothing to do with consciousness or self. This AI is disembodied.
  • The raw data input is (not) curated. It has no added interpretation.
  • Any kind of data can be input. The AI will ignore most of it at first.
  • The AI presents its innards to humans. I am calling these “simulations”.
  • The AI algorithm uses some unspecified form of machine learning.
  • The important innovation here is the ability to generate “simulations”.


The humanist in me says we need to act as the collective Mother for our brain children by providing continual reinforcement for good behavior and discouraging bad behavior. As a world view emerges in the AI, and as an implicit code of morals comes into focus, the AI will “mature”. Widely-expressed fears of AI run amok could be partially alleviated by imposing a Mothering filter on the AI as it comes of age.

Can Anything Evolve without Selection?

I suppose it is possible for an AI to arrive at every possible good idea, insight, and judgement just by digesting the constant data spew from humanity. But without an implicit learning process (such as back-propagation and other feedback mechanisms used in training AI), the AI cannot truly learn in an ecosystem of continual feedback.

Abstract Simulations 

Abstraction in Modernist painting is about generalizing the visual world into forms and colors that substitute detail for overall impressions. Art historians have charted the transition form realism to abstraction – a kind of freeing-up and opening-up of vision.

Imagine now a new path leading from abstraction to realism. And it doesn’t just apply to images: it also applies to audible signals, texts, movements, and patterns of behavior.

Imagine an AI that is set up like the illustration above coming alive for the first time. The inner-life of newborn infant is chaotic, formless, and devoid of meaning, with the exception of reactions to a mother’s smile, her scent, and her breasts.

A newborn AI would produce meaningless simulations. As the first few humans log in to give feedback, they will encounter mostly formless blobs. But eventually, some patterns may emerge – with just enough variation for the human judges to start making selective choices: “this blob is more interesting than that blob”.

As the young but continual stream of raw data accumulates, the AI will start to build impressions and common themes, like what Deep Dream does as it collects images and finds common themes and starts riffing on those themes.

The important thing about this process is that it can self-correct if it starts to veer in an unproductive direction – initially with the guidance of humans and eventually on its own. It also maintains a memory of bad decisions, and failed experiments – which are all a part of growing up.


If this idea is interesting to you, just Google “evolving AI” and you will find many many links on the subject.

As far as my modest proposal: the takeaway I’d like to leave you with is this:

Every brain on earth builds inner-simulations of the world and plays parts of those simulations constantly as a matter of course. The simple animals have extremely simple models of reality. We humans have insanely complex models – which often get us into trouble. Trial simulations generated by an evolving AI would start pretty dumb, but with more sensory exposure, and human guidance, who knows what would emerge!

It would be irresponsible to launch AI programs without mothering. The evolved brains of most complex mammals naturally expect this. Our AI brain children are naturally derived from a mammalian brain. Mothering will allow us to evolve AI systems that don’t turn into evil monsters.

Having sex with robots to save the planet

Long long ago, there was an accident in a warm puddle. A particular molecule – through some chance interaction with the soup of surrounding molecules – ended up with a copy of itself. Since the surrounding soup was similar to the original, the copy was more likely to replicate itself. And so it did. The rest is history. We call it evolution.

It is possible that similar accidents happened elsewhere around the same time – not just in one single puddle. One could also say that variations of this accident are still happening – only now at a massive scale.

Every act of every living thing can be seen as an elaboration of this original act. Self-replication is the original impetus of all life. We share a common ancestor with amoebas – who replicate asexually. The invention of sexual reproduction boosted genetic creativity. More recently in the scope of Earth’s history, creativity escaped the confines of genetics. We humans are the primary hosts of this creative engine.

Human beings have contrived all of the resulting aspects of survival to an art-form. This includes – not just the act of sex – but also the act of preparing food (cuisine), the act of making sounds and speaking (music and singing), and the act of altering the environment to create new structure (visual art). The abstractions and representations of the world that the brain generates via the body are derivations and deviations from the original acts of survival. It’s a form of self-replication.

The emergence of abstractions, mental models, and representations is increasing in complexity. This is an inevitable one-way blossoming accelerated by the emergence of the animal brain. The human experience is conflicted; we are oriented toward achieving escape velocity from Original Nature, but we also long for Original Nature. How can we resolve this conflict?

The original act of self-replication has powerful repercussions – billions of years after the original accident – it has taken on many forms. It is the reason we humans have strange phenomena like orgasm. And selfies.


We are at a crossroads in the history of life on Earth. The current era of global warming is almost certainly the result of the overpopulation and hyperactivity of humans, who have released – and continue to release – too much carbon into the atmosphere. One effective solution to global warming would be to reduce the primary agents of the fever…to reduce human population.

And so, converting that original act of replication into works of art is not just creative and exciting: it may be necessary. Humans must transcend the Earthly act of self-replication in order to preserve the health of the planet.

The future of sex will be…let’s just say…interesting. Every cell in our body contains the blueprint of a desire to replicate. Nature and society are structured around the elaborate machinery that has emerged to ensure self-replication – of human bodies and culture. This desire has made its mark on every aspect of society – even if we don’t recognize it as such. We cannot escape it. And so we need to virtualize it, because self-replication of human beings (physically) has become a threat to the planet that sustains us. It’s our duty to Mother Earth.

I am a living organism and so I have to contend with this crazy desire to replicate. Note: I am childless. I have never replicated my genes and have no intention to do so at this stage in my life. But I am passionate about replicating ideas, art, words, and software.

Now, what about the title of this blog post? Will people eventually start having sex with robots? It will certainly be more subtle than that. In fact, it has been said that by the time we get to that point, WE will be the robots.

Is this the kind of future I want? Strangely, yes. Because I will have long returned to the Earth – my molecules will have been handed down through generations of living things. I will be a part of Earth’s physiology. My tribe will be bigger than humanity.

One of my molecules may even end up in a warm puddle somewhere.

Thoughts on the Evolution of Communication

My dog and I engage in a lot of signaling. But it is not always deliberate, and it is not always conscious, and it is not always a two-way process.

In the morning, Otto licks my bald head. He can probably smell what I have been dreaming. I hold him and we have a nice cuddle. Just one of our many routines. He looks at me and I look at him. He is always checking me out. In the process of getting to know each other over several years we have come to read each other’s signals – our body language, interactions, responses, vocalizations…and smells.

image from

Semiosis emerges in the process. If there is a coupling of signals – a mutually-reinforcing signaling loop – two-way communication emerges. It is not always conscious – for either of us. Sometimes, a mutually-reinforcing signaling process which I was previously unaware of becomes apparent to me. When this happens, I become an active agent in that semiosis.

Otto is so intensely attentive to me – my routines (and deviations from them). He probably tunes-in to many more of my signals than I do to his. But then again, I am a human: I generate a lot of signal. Does he see this as “communication?” It is not clear: his brain is a dog brain, and mine is a human brain. We don’t share the same word for this experience (he only knows a few English words, and “communication” isn’t one of them).

I can be sure of one thing: we share a lot of signaling. And, as members of two highly-social species, we both like that.

I would conclude from this that communication among organisms in general (the biosemiosis that has emerged on Earth over the last few billion years) came about pretty much the same way that Otto and I established our own little world of emergent semiosis. As life evolved, trillions of coupled signaling channels reinforced each other over time and became more elaborate. Eventually, this signaling became conscious and intentional.

And so here we are: human communication has reached a level of sophistication such that I can type these words – and you can read them. And we can share the experience – across time and space.