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.

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.

No Rafi. The brain is not a computer.

Rafi Letzter wrote an article called “If you think your brain is more than a computer, you must accept this fringe idea in physics“.

Screen Shot 2016-06-11 at 12.50.53 PM

The article states the view of computer scientist Scott Aaronson: “…because the brain exists inside the universe, and because computers can simulate the entire universe given enough power, your entire brain can be simulated in a computer.”

Who the fuck said computers can simulate the entire universe?

That is a huge assumption. It’s also wrong.

We need to always look close at the assumptions that people use to build theories. If it can be proven that computers can simulate the entire universe, then this theory will be slightly easier to swallow.

By the way, a computer cannot simulate the entire universe because it would have to simulate itself simulating itself simulating itself.

The human brain is capable of computation, and that’s why humans are able to invent computers.

The very question as to whether the brain “is a computer” is wrong-headed. Does the brain use computation? Of course it does (among other things). Is the brain a computer? Of course it isn’t.

The Singularity is Just One in a Series

I’m reading Kurzweil’s The Singularity is Near.

It occurs to me that the transition that the human race is about to experience is similar to other major transitions that are often described as epochs – paradigm-shifts – in which a new structure emerges over a previous structure. There are six key epochs that Kurzweil describes. (The first four are not unlike epochal stages described by Terrance Deacon and others.)

  1. Physics and Chemistry
  2. Biology and DNA
  3. Brains
  4. Technology
  5. Human Intelligence Merges with Human Technology
  6. Cosmic Intelligence

When a new epoch comes into being, the agents of that new epoch don’t necessarily eradicate, overcome, usurp, reduce, or impede the agents of the previous epoch. Every epoch stands on the shoulders of the last epoch.This is one reason not to fear the Singularity…as if it is going to destroy us or render us un-human. In fact, epoch number 5 may allow us to become more human (a characterization that we could only truly make after the fact – not from our current vantage point).

I like to think of “human” as a verb: as a shift from animal to post-human, because it characterizes our nature of always striving for something more.

animal to posthuman

There are debates raging on whether the Singularity is good or bad for humanity. One way to avoid endless debate is to do the existential act: to make an attempt at determining the fate of humanity, rather than sit passively and make predictions.  As Alan Kay famously said, “the best way to predict the future is to invent it”. We should try to guide the direction of the next epoch as much as we can while we are still the ones in charge.

In a previous article I wrote that criticizes some predictions by Nick Bostrom, I compare our upcoming epochal shift to a shift that happened in the past, when multi-cellular beings evolved. Consider:

Maybe Our AI Will Evolve to Protect Us And the Planet

tree-of-lifeBillions of years ago, single cells decided to come together in order to make bodies, so they could do more using teamwork. Some of these cells were probably worried about the bodies “taking over”. And oh did they! But, these bodies also did their little cells a favor: they kept them alive and provided them with nutrition. Win-win baby!

I am not a full-fledged Singularitarian. I prefer to stay agnostic as long as I can. Its not just a human story. Our Singularity is just the one that is happening to us at the moment.

Similarly, the emergence of previous epochs may have been experienced as Singularities to those that came before.

The Evolution of Mathematics on Planet Earth

tumblr_ncxcqixe9k1qzy92fo1_1280

math-heartMany people couldn’t imagine Math and Biology going out on a date. Flirting with each other from time to time…maybe. But a date? Never! Math is precise, abstract, cool, and distant. Biology is messy, unpredictable, prone to mood swings, and chemically dependent…as it were.

But this may be changing.

“The conversion of biology into a more quantifiable science will continue to the extent that it might even become the main driving force behind innovation and development in mathematics”

Philip Hunter

Let me explain why I think Math and Biology are ultimately compatible, and in fact, part of a Single Reality.

tumblr_my165xRv5f1swaxzgo1_250

I have written a few articles on the subject of math, and raised questions as to the universality, truth-status, and God-givenness of Math. Here is something to consider about Math and Biology:

Math Evolved in the Biosphere

Let’s start with numbers. Imagine a mother crow busily feeding her three chicks. She would become worried if she came back to her nest to suddenly find two chicks instead of three.

House_Crow_feeding_chicks

She would know there something is wrong with this picture…because crows can count (they can subitize small numbers, like about 2 or 3).

How did it come about that some animals, like crows and humans, can count? First of all, in order for intelligent beings to be able to count, they have to live in an environment where countable objects are found, and where counting has some evolutionary benefit. Consider a gaseous planet where fluids intermix and there is no way to detect a “thing” or “event” and to compare that with another “thing” or “event”. In this kind of world, there is nothing to count.

seahorseFor that matter, it is unlikely that an intelligent entity that can count could ever evolve on such a planet in the first place, because structure and differentiation at some physical level are required for living things to bootstrap themselves into existence.

Theories of autopoiesis, negentropy, and the emergence of mind from matter rely on the existence of a prior structure to the universe where it is possible for self-regulation, and self-creation to arise. One might say that the origins of life had a head start long before those first molecules started dancing together and accidentally reproducing. Maybe it wasn’t such an accident after all.

imgres

…which brings me to a core concept: since Earth’s biosphere gave rise to animals that can count, as well as those things that can be counted – at the same time, we must understand ourselves as in and of the biosphere – we and it all evolved together: one did not come before the other.

70212-1024x603Which came first: the chicken or the egg? Neither. They have both been in a continual state of becoming since egg-like things and chicken-like things have existed. And if you go back in time far enough, these things look less and less like chickens and eggs.

We animals have evolved to understand containment, and that is partly because hierarchy evolved within the fabric of physical biology. We know what it means for something to be “inside” or “outside” of something else. We clumpify, categorize, differentiate, compare, and identify. All animals need some degree of this compartmentalization of nature in order to operate within it.

We cannot separate our math from the environment from which it evolved. The very foundations of math evolved within the bodies and minds of animals as a part of evolution. At least this is what several recent scientists and philosophers are suggesting. (Mathematicians are more likely to claim that math is universal, constant, and unchanged by biology.)

OctoMath

In a previous article I consider what kind of math would have emerged if octopuses has evolved to become the complex and dominant species on earth, instead of humans. This is not so hard to imagine, considering how intelligent they are.

Screen Shot 2015-08-10 at 12.19.31 AM

Would an advanced octopus race have stumbled upon complex numbers? Would they have become as obsessed with the Cartesian coordinate system as we are? Since they have no skeletons, would they have formulated a geometry based on angles and lengths? Of course we can’t know, but it is likely that they would have created some math concepts that we may never achieve. And that would be because the long history of math that we have built and that we rely on to create new math has taken our brains and societies too far away from the place where an octopus-like math would naturally arise.

mouroborobius2Now consider aliens from a completely different kind of planet than Earth. What kind of math would originate in that world? Many people would argue that math is math and it doesn’t matter who or what discovers or articulates it. And there may be some truth to this. But we can only hope and imagine that this is the case.

Until we meet aliens from another planet and ask them if they understand and appreciate the fibonacci sequence, I have to assume that their math is different than ours.

What do you think?

(I would have consulted one of my octopus friends on the subject…but I don’t speak their language).

The Future CAN Change the Present (Terrence Deacon’s Incomplete Nature)

DIn his book, Incomplete Nature – How Mind Emerged from Matter, Terrence Deacon tackles some of the deepest and gnarliest philosophical questions about life and mind, and how they emerge from the physical world despite the fact that the laws of physics can only explain lifeless, mechanical processes. Regarding the second law of thermodynamics, which states that everything is in an eternal state of eventual decay, Deacon says this:

220px-TerryDeacon“There is the dead, pointless, uncaring world and its rules, and the living, striving, feeling world and its rules, and the two seem to be working in quite contradictory ways. Because the spontaneous order generation that is so characteristic of life and mind runs counter to this otherwise exceptionless current of nature, it demands that we take seriously the possibility that our usual forms of explanation might be inadequate. When unrealized future possibilities appear to be the organizers of antecedent processes that tend to bring them into existence, it forces us to look more deeply into the ways we conceive of causality and worry that we might be missing something important.”

I am in the process of re-reading the book, which is fat and dense.

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Deacon makes little attempt to help the reader by phrasing his sentences and paragraphs for easy digestion. Subsections are rare, and I feel like I’m often wading in a vast ocean without a view of an outcropping to pause and catch my breath. I beg for a period, perchance a comma. But despite the relentless texture of the writing, what Deacon has to say is inspiring…almost breathtaking. Actually, the breath-taking happens after I allow Deacon’s ideas to percolate in my mind during a short walk – which I find necessary for getting the depleted oxygen back into my brain.

…which reminds me of a core concept of the book. Billions of years of evolution have gone into the making of this unlikely spacetime dynamical event, which is me: a fully conscious human who happens to be writing words for you to read at some point in the future. The fact that I can choose to go for a walk for mental and physical health, and maintain my equilibrium (on many levels of consciousness, mostly un), is nothing short of a miracle. This is especially true when you consider the general trajectory of atoms in the universe.

31d377bBut in fact it is not a miracle. It can be explained with a systems-attitude overlaid on top of physics that includes emergence. “Emergence” is so wonderfully explained by Daniel Dennett in his descriptions of skyhooks and cranes. Here’s my take on the topic: when a higher level of structure emerges in the world, it reaches down to manipulate the lower levels that brought it into existence – it regulates and modulates those lower levels for its own survival. Then another level of emergence comes into being, and begins to regulate and modulate the previous structures. This upward cascading of order – this ever-complexifying hierarchy of constraints – is particular to life and mind, and it runs counter to the entropy that is the general rule of the universe. This is sometimes referred to as negentropy.

fdd042bb5ac4e50236974f747f0fdb8cI had previously struggled to understand the “incompleteness” aspect of Deacon’s thesis. In the first part of the book, he spends a lot of time re-stating the notion of things that “aren’t there” or that exist “for the sake of something missing”, after only briefly defining this concept up-front. In my first reading I had started out with a limp. Perhaps some training wheels with familiar language would have helped, as these are new and subtle ideas.

This is not unlike a small criticism from Daniel Dennett, in a book review. Overall though, Dennett’s review is very positive. In fact, Dennett says the book has him re-examining his fundamental working assumptions. When something rocks Daniel Dennett’s world, you stop and take notice.

The Future Does Change the Present

illusionLiving things have inner representations of the world they find themselves in – that’s part of the holographic picture of life. For us humans, these representations are not just in our brains; they are infused in our technology – distributed throughout our extended phenotypes. Organisms with minds not only put a lot of effort in trying to predict the future (which is necessary for survival), but they act in accordance to those predictions.

This has the effect of creating alternate futures – of changing the course of events – based on something represented in the mind of the organism – represented but not actually “there” (though it could be there in the future). I believe this is what Deacon means by incompleteness, things that are missing, or, in his own words: “absential features“.

RV-AE885_BRAIN2_DV_20111111015014According to Raymond Tallis in a Wall Street Journal book review, An absential is a phenomenon “whose existence is determined with respect to an . . . absence.” This sounds somewhat opaque but captures something essential to mind. In the push-pull universe of mechanical causation, only that which is present shapes the course of events. In our lives, by contrast, we are always taking account of things that are no longer present or not yet present or that may never come to pass. Thus “absentials” include our beliefs, the norms to which we subscribe and those great silos of possibility such as “tomorrow” and “next year.”

Screen Shot 2015-07-25 at 9.30.00 PMAlan Kay is once said, “The best way to predict the future is to invent it“. This is not just good advice for entrepreneurs; it might also be the motto for all living things. How else can we explain the fact that life on Earth has continued to exist for billions of years, with robust self-similar structures that persist for several millions of years – gradually changing, only to become better at predicting the future.

Why do we not expect every living thing to quickly dissolve into a gas, or at least some entropic state of uselessness? This is what classical physics would predict for such a rare, unlikely, and delicate assemblage of molecules. The answer is that living things have their futures infused into their DNA, their minds, and their societies. We fight the force of entropy, and we succeed.

For the most part.

Physics does not (at least not yet) adequately explain how the future effects the present. For this reason, Deacon is helping to build the foundations for a new kind of physics (or perhaps a new kind of meta-physics), which includes…

you, me, our consciousness, and everything that has meaning.