Very large numbers are not numbers: Infinity does not exist

(this blog post was originally published in https://eyemath.wordpress.com/ . 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.

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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:

80658175170943878571660636856403766975289505440883277824000000000000

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.

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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.

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.

Warming

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 http://projectdolittle.com/

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.

The Information EVOLUTION

I remember several decades ago learning that we were at the beginning of an information revolution. The idea, as I understood it, was that many things are moving towards a digital economy; even wars will become information-based.

The information revolution takes over where the industrial revolution left off.

I am seeing an even bigger picture emerging – it is consistent with the evolution of the universe and Earth’s biosphere.

Screen Shot 2016-06-16 at 10.20.47 AM

At the moment, I can hear a bird of prey (I think it’s a falcon) that comes around this neighborhood every year about this time and makes its call from the tree tops. When I think about the amount of effort that birds make to produce mating calls, and other kinds of communication, I am reminded of how much importance information plays in the biological world. The variety and vigor of bird song is amazing. From an evolutionary point of view, one has to assume that there is great selective pressure to create such energy in organized sound.

money+gorilla+teeth+omg+weird+primatesThis is just a speck of dust in comparison to the evolution of communication in our own species, for whom information is a major driver in our activities. Our faces have evolved to give and receive a very high bandwidth of information between each other (Compare the faces of primates to those of less complex animals and notice the degree to which the face is optimized for giving and receiving information).

Our brains have grown to massive proportions (relatively-speaking) to account for the role that information plays in the way our species survives on the planet.

Now: onto the future of information…

Beaming New Parts to the Space Station

Screen Shot 2016-06-16 at 10.29.58 AM

Guess which is more expensive:

  1. Sending a rocket to the space station with a new part to repair an old one.
  2. Beaming up the instructions to build the part on an on-board 3D printer.

You guessed it.

And this is where some people see society going in general. 3D printing will revolutionize society in a big way. Less moving atoms, More moving bits.

To what degree will the manipulation of bits become more important than the manipulation of atoms?

Not Just a Revolution: Evolution

My sense is that the information revolution is not merely one in a series of human eras: it is the overall trend of life on Earth. We humans are the agents of the latest push in this overall trend.

Some futurists predict that nanotechnology will make it possible to infuse information processing into materials, giving rise to programmable matter. Ray Kurzweil predicts that the deep nano-mingling of matter and information will be the basis for a super-intelligence that can spread throughout the universe.

Okay, whatever.

For now, let’s ride this information wave and try to use the weightlessness of bits to make life better for all people (and all life-forms) on Earth – not just a powerful few.

When Earth Discovered Water

When Earth discovered water, life became possible.

header-earth-from-space

“Discover”? I was going to use the word “invent”. Then I remembered that water can be found on other planets, comets, asteroids…and the Moon. Water is not unique to Earth. It may be more accurate to say that Earth invented a way to preserve and manage its water by evolving the biosphere. The biosphere harnesses, protects, filters, and enlivens the water that covers most of the planet.

Mars did not succeed in preserving its water.

The Gaia Hypothesis blows open the perspective of what life is. The evolution of the self-adaptive, self-regulating spherical ecosystem that we call Earth is more than just a collection of interacting organisms. It also relies on the dynamics of storms, oceans, tectonic plates, and the balance of gases in the atmosphere.

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I began thinking about this as I was pouring boiling water into a coffee press. A beautiful stream, sparkling and transparent, visible only by virtue of the fact that it reflects the darks and lights of the surrounding environment.

What an amazing fluid. A true friend of gravity and heat, forming a collaboration resulting in a network of clouds, rain, snow, glaciers, streams, rivers and oceans.

There’s a deep reason why we can generate so much poetry about water.

“Planetary scientists are quick to stress that it’s not just water that’s indispensable for life, but liquid water. The distinction is key”

Water made life on Earth possible. Some people go as far as to say that water is alive.

Screen Shot 2016-05-16 at 12.32.00 PM

On the Origins of Earth’s Water

http://www.iflscience.com/physics/origin-earths-water-discovered-0

Did life on Earth begin with replicating molecules? Depends on how you define life. The backdrop for these replicating molecules was already rich and dynamic, with an interplay of water (in all its forms), gravity, atmospheric and ocean chemistry, electric storms…

…and the rhythms of day and night, winter and summer, which forms a backbeat. That backbeat drives the polyrhythmic dance that pulls water through its many forms – and brings us into being.

early-earth-moon-system-closeup

 

Thoughts on the Evolution of Evolvability

Evolve-Darwin-Fish-Car-Emblem-(2363)

It is early February. The other day, I observed some fresh buds on a tree. When I lived back east, I remember seeing buds on bare trees in the snowy dead of winter. I used to wonder if these trees are “preparing” for the first days of spring by starting the growth of their buds. Trees, like most plants, can adapt to variations of weather. All organisms, in fact, exhibit behaviors that appear resourceful, reactive, adaptive, even “intelligent”.

We sometimes talk about animals and plants in terms of their goals and intentions. We even use intentional language in relation to computers or mechanical machines. Even though we know a machine isn’t alive, we use this kind of language as a form of shorthand.

But there may be something more than just verbal shorthand going on here.

The Intentional Stance

Daniel Dennett proposed the concept of the Intentional Stance. When I first learned about this idea, I felt a new sense of how our own human intelligence is just a special case of the adaptive and goal-directed nature of all life on the planet.

When I saw those buds on the tree the other day, I realized that there is so much goal-directed behavior happening all over the place – in plants, animals, and even in ecological systems. Are humans any more adaptive or “intentional” than any other organism?

The Evolution of Self and the Intentional Stance

Could it be that our human brains have simply…

…wrapped a fully-evolved self around our intentions?

…that we are really no more goal-directed or intentional than any other organism…except that we reflect on it with a higher level of consciousness, and apply a fully-formed language to that intentionality?

The Evolution of Evolvability

I first learned of the evolution of evolvability from a paper by Richard Dawkins. It’s a powerful idea, and it helps to make evolution seem less magical and perhaps easier to imagine. Not only have organisms continued to evolve, but their ability to evolve has improved. An example is the evolution of sexual reproduction, which created a huge advantage in a species’ ability to exploit genetic variation over evolutionary time.

A recent article titled “Intelligent design without a creator? Why evolution may be smarter than we thought” makes reference to the Evolution of Evolvability. It helps to cast the notion of intelligence and learning as prolific and pervasive in the natural world.

It would appear that the ability to evolve better ways to evolve predates humans. (It might even predate biology).

Of course we humans have found even better ways to evolve – including ways that overtake or sidestep our own human biology. This constitutes a new era in the evolution of life on earth – an era in which technology, culture, and ideas (memes) become the primary evolving agents of our species (and possibly the whole planet – assuming we humans make the planet so sick that we have to fabricate artificial immune systems in order to keep the planet (and thus ourselves) healthy.

While many people will cast this Singularity-like idea in a negative light, I see it as a new protective organ that is forming around our planet. Biology is not going away. It is just one regime in a progression of many emergent regimes. Biology has given birth to the next regime (via Dennett’s crane), which then reaches down to regulate, modulate, and protect the regime which created it.

Evolvability is the higher-level emergent system over evolution. It is a higher-order derivative. When seen in this way, biology comes out looking like just one step in a long process.

(Thanks to Stephen Brown for editorial assistance)