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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Number Portraits

I designed a set of number portraits for the integers from 1 to 36.

Each number is either a prime or a composite. If it is a prime number, then it has two divisors: 1 and itself. This is visualized as the gray-colored half circle, where the top represents 1 and the bottom represents the number itself. Composite numbers have other pairs of divisors, and these are visualized as the smaller, colored arcs.

The perfect squares (4, 9, 16, 25, and 36) each have a line segment located at the square root.

The numbers 6 and 8 each have one pair of divisors (besides 1 and themselves); they are (2,3) and (2,4), respectfully. Since the first number in each pair is 2, these arcs are colored green. Divisor pairs in which the first divisor is 3 are colored blue; divisor pairs in which the first divisor is 4 are colored red. And divisor pairs in which the first divisor is 5 are colored yellow. These colors visualize divisibility by these first 5 positive integers.

The highly-composite numbers 12, 24, and 36 are shown enlarged below.

The number 36 has the most divisors in this set. It’s divisor pairs are

(1, 36)  (2, 18) (3, 12) (4, 9) (6, 6).

These are in a similar spirit to the Divisor Plot images I created in 2010:

http://www.divisorplot.com/index.html

 

Thelonius Monk’s Shapeshifting Chord

One of my part-time hobbies is being a Monk interpreter. A Monk interpreter not only learns how to play Monk’s compositions, but also makes a point of getting into the head of this eccentric man. The reason to do this is that Monk was an improvisor – and he was driven by an inner vision. If you can tap that inner vision, then you can generate Monk-like music – and improvise on it…even while playing Beatles songs.

I wrote a piece in 2013 about Monk as a mathematician.

Math can be about patterns (visual or sonic). Math does not always have to be expressed in numbers. Monk once said,“All musicians are subconsciously mathematicians”.

A Symmetrical Chord

The chord I’m talking about has four notes. It is typically used as a dominant chord – which naturally resolves to the tonic. Unlike the classical dominant-seventh, this chord has a flatted fifth – which makes it slip into a symmetrical regime – as shown in the picture above – inscribed in the circle of fifths.

 

According to Wikipedia, this chord is called the “Dominant Seventh Flat-Five Chord“. The cool trick about this chord is that it can resolve to either of two different tonics – each being a tri-tone apart.

So for instance, a chord with these notes:     Eb   F   A   B      can resolve to either Bb or E as the home key.

This chord also happens to contain 4 of the 6 tones in a whole tone scale, which Monk famously used (often as a dominant arpeggio).

If you are not familiar with music theory, you may still appreciate the beauty of sonic geometry and how it can generate such variety. If you apply similar concepts to rhythm as to harmony then you have a wonderfully rich canvas for endless musical expression. I like the way Monk wove these geometries together in a way that makes the foot tap and the ear twinge – and the brain tweak.

Monk was of course not the only one to apply these ideas – but he did accomplish something remarkable: the application of embodied math. If you have spent as much time as I have learning his language, listening to him improvise can cause a smile – or the occasional giggle – to pop out. Like an inside joke.

There is plenty of material on the internet about Monk. Here’s one voice among the many who have acquired an appreciation for Monk:  How to Listen to Thelonius Monk – by George H. Jensen, Jr.

Enough with this Square Root of -1 Business!

Like so many other people, I was kept from appreciating the beauty and utility of mathematics because of the way it was taught to me.

The majority of introductions to complex numbers start with the elusive and mysterious square root of -1, denoted by i.

A number that has an i stuck on to it is called “imaginary” (a convenient differentiator to “real”). Being asked to learn something that is called “imaginary” is not very motivating to young learners who work best starting with concrete metaphors.

The imaginary number i is counterintuitive and confusing. And it’s not the coolest part. Sure, i was an important invention at a critical stage in the history of math when there was no good way to express z² = -1. And yes, it makes a good ending to a long story (which happens to be true): math has advanced through several expansions of the concept of “number” … from the counting numbers to the wholes – to the negatives – to the fractions – to the irrationals – and finally to complex numbers – where i came along and saved the day.

But…does this mean that invoking i is the best way to explain complex numbers to novices – to everyday people? I join many others in saying that there is a better way to learn about the wonderful world of two-dimensional numbers. One voice among those is Kalid Azad.

He speaks in metaphors and freely engages the visual mind to help us grasp math concepts using our whole brain. In his explanation on complex numbers, Azad says this about i: “It doesn’t make sense yet, but hang in there. By the end we’ll hunt down i and put it in a headlock, instead of the reverse.”

…..

When you get an intuitive, aesthetic feeling for why certain mathematical ideas are being taught, you become more motivated to learn the notation. The corollary: learning math notation without understanding why is like learning musical notation before ever being allowed to listen to or play music.

Paul Lockhart, in A Mathematician’s Lament, compares the way math is taught to a nightmare scenario in which music is taught to students using sheet music notation only (no actual music is played or heard) – until the student is advanced enough to start “using” it.

 

What is a Two-Dimensional Number?

When I read that complex numbers are really no more “imaginary” than real numbers, I decided that I would start dismantling my old worldview. Why should I assume that numbers have to be one-dimensional? Over time, I became more accustomed to the notion that a number can occupy a plane (the complex plane) and not just a line (the number line). Learning how to make images of the Mandelbrot Set helped a lot.

Think of Multiplication as Rotation

Instead of trying to wrap your mind around i, and how it magically makes equations come out right, let’s start with geometry. Think of multiplication as rotation and expansion. In the blog Girls Angle, Ken Fan introduces complex number multiplication in a nice visual way… here.

Here’s a video explaining complex numbers in terms of physical metaphors, and eventually explaining why the square root of -1 becomes a necessary part of the notation.

Squaring

Consider the following diagram showing what happens when you square certain complex numbers that lie on the unit circle:

The dot on the right represents the complex number (1+0i). When you square it, it stays the same (no surprise: 1×1=1). The number at the left is (-1+0i). When you square that, it becomes (1+0i). But when you square the number at top (0+1i) it “rotates” by 90 degrees to (-1+0i). Finally, at the bottom, the number (0-1i) rotates…but would it be correct to say that it rotates by 90 degrees clockwise to (-1,0i)? Depends on how you look at it. Rotating by 270 degrees counter-clockwise has the same result. This is the nature of rotation and angular reality: it is periodic – it cycles…it repeats.

What an awesome idea. Multiplication is like doing a whirling dervish jig.

Animated Squaring

Here’s an interactive tool I made that allows you to play with 200 dots (complex numbers) randomly scattered on the complex plane. You can experience what happens when complex numbers are squared. It also allows you to multiply the dots (using a complex number dot that you can drag along the screen).

http://ventrella.com/ComplexSquaring/

This interactive tool might make you feel as if the dots on the screen are obeying some sort of gravitational law of physics. Well, in a way, yes, that’s what’s happening. When you add, multiply, or exponentiate numbers, you get a new number. In the complex plane, the space where that change takes place is two-dimensional. That’s cool! We like images.

Here’s another visual tool: when we multiply two complex numbers, such as (a+bi) and (c+di), we can visualize the operation in this way:

In pseudocode:

realPart      = (a*c) - (b*d);
imaginaryPart = (a*d) + (b*c);


This explanation of multiplication does not require i.

To this day, I STILL do not feel very much music when I think about the square root of -1.

On the other hand, the more I play around with visualizing and animating complex numbers, the more intuitive they become, and the deeper my sense that these numbers are as real as any old one-dimensional number.

They are not imaginary at all.

Math Word Problems are Problematic

Mark Twain said: “Never let school interfere with your education.“

Here’s a math riddle:

“Peter has 21 fewer marbles than Nancy. If Peter has 43 marbles, how many marbles does Nancy have?”

The first sentence requires me to do some linguistic fiddling. There is an implication that both Peter and Nancy possess marbles – but it is not directly stated. The second sentence begins with “If”, which means the primary grammatical elements in the question are postponed until the end. Let’s re-phrase this riddle to say:

“Peter and Nancy each have a bag of marbles. Peter has 43 marbles in his bag. Peter has 21 fewer marbles than Nancy has. How many marbles does Nancy have in her bag?”

….

This might make the riddle easier to solve. Or it might not. Either way, I can say for sure that all this wordy bullshit is irrelevant to the actual math.

Math, like Music, is a Universal Language

Now consider what it would be like if you were naturally talented in math, and you were faced with a math riddle expressed in English…but English were not your first language. You may have to spend more time on the question, and you may make some critical mistakes. The subtleties of one language may not translate to another language, causing you to trip up.

We are playing with words here. Now, playing with words is fine; it’s part of how we learn to speak, listen, read, and write. In fact, playing with words that have mathematical content is a good exercise. But this should not come into play for testing students on math skills. The problem (as always) is in the testing.

Here’s another one:

“Sue has two pencils. She spends one hour at the store and buys three more pencils. How many pencils does Sue have in all”.

WTF does “spends one hour in the store” mean? Is this just narrative fluff, or is there some clever hint in there?

If I had been presented with this problem as a young student, I would have spent some time mulling over “spends one hour at the store”. However, this is irrelevant and unrelated to the answer.

How to Obfuscate Mathematical Thinking With Clever Language

For dyslexic students, students who learn through action (kinesthetic learners), students who are visual thinkers, and students who learn best by building things, this wordsmithing can be a recipe for failure.

In the real world of adults getting things done and making a living, math is rarely experienced in the form of clever riddles. Math – at its best – is manifested deep within the texture of our daily actions.

Here’s another one:

“You have 24 cookies and want to share them equally with 6 people. How many cookies would each person get?”

Let’s think about this. I “have” 24 cookies. (That’s a lot of cookies – why would I have so many cookies?) I “want” to share them with 6 people. Okay. I have a desire to share cookies. So far so good. I’m a generous guy! But then the second sentence appears unrelated: “How many cookies would each person get?” Wait a minute: am I about to give these cookies to these people? And what exactly does “equally” mean?

I know it may seem trivial for me to analyze these details. As an adult I know what this sentence means. But as a young student, I may not have had the full vocabulary or grammatical wherewithal to jump right to an answer. Also, as a “narrative learner”, I would have really wanted to make sure I understood the characters involved, their motivations, etc. I could imagine getting easily get swept up by the storyline (simple as it is).

In short, by working out the characters of this story and their motivations, I may not actually be doing math: I might be engaging in language craft and storytelling. Which is great! But this should not interfere with my being tested on my innate math skills.

Here’s another:

“Kennedy had 10 apples. She gave some to John. Now she has 2 apples left. How many apples did she give to John?”

The tense of this little story jumps back and forth between past and present. At age 55, I am now quite facile with language, but when I was 10, I would have had to put in some effort into parsing these shifts in tense.

In fact, my language skills were quite poor when I was 10, and this had an impact on all my school subjects (not just math). Later in life, after I had escaped school and actually started to gain some relevant skills, MIT offered me an opportunity to earn a Master’s degree. They did not ask me any math riddles. MIT knows better than that.

Language

One might argue that language skills are fundamental and important for learning most anything. That’s accurate. Reading, writing, speaking, and listening are fundamentally useful, and the better you are at language, the better you are likely to become at most other skills.

If this is the case, we might conclude that mixing grammatical sentence structure with mathematical logic is a valuable skill.

Indeed.

But school curriculum designers should not confuse the ability to parse a cleverly-crafted sentence with one’s innate mathematical abilities.

The problem, as always, is with TESTING.

I’ll close with this:

An Open Letter to the Education System: Please Stop Destroying Students

Why Nick Bostrom is Wrong About the Dangers of Artificial Intelligence

Nick Bostrom is a philosopher who is known for his work on the dangers of AI in the future. Many other notable people, including Stephen Hawking, Elon Musk, and Bill Gates, have commented on the existential threats posed by a future AI. This is an important subject to discuss, but I believe that there are many careless assumptions being made as far as what AI actually is, and what it will become.

Yea yea, there’s Terminator, Her, Ex Machinima, and so many other science fiction films that touch upon deep and relevant themes about our relationship with autonomous technology. Good stuff to think about (and entertaining). But AI is much more boring than what we see in the movies. AI can be found distributed in little bits and pieces in cars, mobile phones, social media sites, hospitals…just about anywhere that software can run and where people need some help making decisions or getting new ideas.

John McCarthy, who coined the term “Artificial Intelligence” in 1956, said something that is totally relevant today: “as soon as it works, no one calls it AI anymore.” Given how poorly-defined AI is – how the definition of it seems to morph so easily, it is curious how excited some people get about its existential dangers. Perhaps these people are afraid of AI precisely because they do not know what it is.

Elon Musk, who warns us of the dangers of AI, was asked the following question by Walter Isaacson: “Do you think you maybe read too much science fiction?” To which Musk replied:

“Yes, that’s possible”….“Probably.”

Should We Be Terrified?

In an article with the very subtle title, “You Should Be Terrified of Superintelligent Machines“, Bostrom says this:

“An AI whose sole final goal is to count the grains of sand on Boracay would care instrumentally about its own survival in order to accomplish this.”

Point taken. If we built an intelligent machine to do that, we might get what we asked for. Fifty years later we might be telling it, “we were just kidding! It was a joke. Hahahah. Please stop now. Please?” It will push us out of the way and keep counting…and it just might kill us if we try to stop it.

Part of Bostrom’s argument is that if we build machines to achieve goals in the future, then these machines will “want” to survive in order to achieve those goals.

“Want?”

Bostrom warns against anthropomorphizing AI. Amen! In a TED Talk, he even shows a picture of the typical scary AI robot – like so many that have been polluting the air waves of late. He discounts this as anthropomorphizing AI.

And yet Bostrom frequently refers to what an AI “wants” to do, the AI’s “preferences”, “goals”, even “values”. How can anyone be certain that an AI can have what we call “values” in any way that we can recognize as such? In other words, are we able to talk about “values” in any other context than a human one?

From my experience in developing AI-related code for the past 20 years, I can say this with some confidence: it is senseless to talk about software having anything like “values”. By the time something vaguely resembling “value” emerges in AI-driven technology, humans will be so intertwingled with it that they will not be able to separate themselves from it.

It will not be easy – or possible – to distinguish our values from “its” values. In fact, it is quite possible that we won’t refer to it at “it”. “It” will be “us”.

Bostrom’s fear sounds like fear of the Other.

That Disembodied Thing Again

Let’s step out of the ivory tower for a moment. I want to know how that AI machine on Boracay is going to actually go about counting grains of sand.

Many people who talk about AI refer to many amazing physical feats that an AI would supposedly be able to accomplish. But they often leave out the part about “how” this is done. We cannot separate the AI (running software) from the physical machinery that has an effect on the world – any more than we can talk about what a brain can do that has been taken out one’s head and placed on a table.

It can jiggle. That’s about it.

Once again, the Cartesian separation of mind and body rears its ugly head – as it were – and deludes people into thinking that they can talk about intelligence in the absence of a physical body. Intelligence doesn’t exist outside of its physical manifestation. Can’t happen. Never has happened. Never will happen.

Ray Kurzweil predicted that by 2023 a $1,000 laptop would have the computing power and storage capacity of a human brain. When put in these terms, it sounds quite plausible. But if you were to extrapolate that to make the assumption that a laptop in 2023 will be “intelligent” you would be making a mistake.

Many people who talk about AI make reference to computational speed and bandwidth. Kurzweil helped to popularize a trend for plotting computer performance along with with human intelligence, which perpetuates computationalism. Your brain doesn’t just run on electricity: synapse behavior is electrochemical. Your brain is soaking in chemicals provided by this thing called the bloodstream – and these chemicals have a lot to do with desire and value. And… surprise! Your body is soaking in these same chemicals.

Intelligence resides in the bodymind. Always has, always will.

So, when there’s lot of talk about AI and hardly any mention of the physical technology that actually does something, you should be skeptical.

Bostrom asks: when will we have achieved human-level machine intelligence? And he defines this as the ability “to perform almost any job at least as well as a human”.

I wonder if his list of jobs includes this:

Intelligence is Multi-Multi-Multi-Dimensional

Bostrom plots a one-dimensional line which includes a mouse, a chimp, a stupid human, and a smart human. And he considers how AI is traveling along this line, and how it will fly past humans.

Intelligence is not one dimensional. It’s already a bit of a simplification to plot mice and chimps on the same line – as if there were some single number that you could extract from each and compute which is greater.

Charles Darwin once said: “It is not the strongest of the species that survives, nor the most intelligent that survives. It is the one that is most adaptable to change.”

Is a bat smarter than a mouse? Bats are blind (dumber?) but their sense of echolocation is miraculous (smarter?)

Is an autistic savant who can compose complicated algorithms but can’t hold a conversation smarter than a charismatic but dyslexic soccer coach who inspires kids to be their best? Intelligence is not one-dimensional, and this is ESPECIALLY true when comparing AI to humans. Plotting them both on a single one-dimensional line is not just an oversimplification. By plotting AI on the same line as human intelligence, Bostrom is committing anthropomorphism.

AI cannot be compared apples-to-apples to human intelligence because it emerges from human intelligence. Emergent phenomena by their nature operate on a different plane than what they emerge from.

WE HAVE ONLY OURSELVES TO FEAR BECAUSE WE ARE INSEPARABLE FROM OUR AI

We and our AI grow together, side by side. AI evolves with us, for us, in us. It will change us as much as we change it. This is the posthuman condition. You probably have a smart phone (you might even be reading this article on it). Can you imagine what life was like before the internet? For half of my life, there was no internet, and yet I can’t imagine not having the internet as a part of my brain. And I mean that literally. If you think this is far-reaching, just wait another 5 years. Our reliance on the internet, self-driving cars, automated this, automated that, will increase beyond our imaginations.

Posthumanism is pulling us into the future. That train has left the station.

But…all these technologies that are so folded-in to our daily lives are primarily about enhancing our own abilities. They are not about becoming conscious or having “values”. For the most part, the AI that is growing around us is highly-distributed, and highly-integrated with our activities – OUR values.

I predict that Siri will not turn into a conscious being with morals, emotions, and selfish ambitions…although others are not quite so sure. Okay – I take it back; Siri might have a bit of a bias towards Apple, Inc. Ya think?

Giant Killer Robots

There is one important caveat to my argument. Even though I believe that the future of AI will not be characterized by a frightening army of robots with agendas, we could potentially face a real threat: if military robots that are ordered to kill and destroy – and use AI and sophisticated sensor fusion to outsmart their foes – were to get out of hand, then things could get ugly.

But with the exception of weapon-based AI that is housed in autonomous mobile robots, the future of AI will be mostly custodial, highly distributed, and integrated with our own lives; our clothes, houses, cars, and communications. We will not be able to separate it from ourselves – increasingly over time. We won’t see it as “other” – we might just see ourselves as having more abilities than we did before.

Those abilities could include a better capacity to kill each other, but also a better capacity to compose music, build sustainable cities, educate kids, and nurture the environment.

If my interpretation is correct, then Bolstrom’s alarm bells might be better aimed at ourselves. And in that case, what’s new? We have always had the capacity to create love and beauty … and death and destruction.

To quote David Byrne: “Same as it ever was”.

Maybe Our AI Will Evolve to Protect Us And the Planet

Here’s a more positive future to contemplate:

AI will not become more human-like – which is analogous to how the body of an animal does not look like the cells that it is made of.

Billions 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!

To conclude, I disagree with Bostrom: we should not be terrified.

Terror is counter-productive to human progress.

The Evolution of Mathematics on Planet Earth

Many 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.

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.

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.

For 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.

…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.

Which 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.

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.

Now 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 Beauty of Gray Code …………..

http://www.mathworks.com/matlabcentral/fileexchange/40928-generate-gray-code-disk

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http://anthony.liekens.net/index.php/Misc/TrueBinaryTime

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http://vision.middlebury.edu/~schar/papers/structlight/p1.html

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 http://www.jeffreythompson.org/blog/tag/gray-code/

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http://www.fachlexika.de/technik/mechatronik/sensor.html

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Gray Code is an alternative binary representation, cleverly devised so that, between any two adjacent numbers, only one bit changes at a time. If there is an error reading any bit that has changed then, at worse, the read value will never be out by more than one unit.

This has tremendous value in the real world. Computers might be digital, but we live in an analog world. Interfaces between these need to be carefully considered.

http://www.qsl.net/oe5jfl/encoder.htm

Our Colorful Mathematics Revolution

Education bureaucrats are trying to gently and safely tweak a broken system so that fewer students fail math.

Meanwhile, a colorful revolution is taking shape outside the walls of a crumbling institution. A populist movement in creative math is empowering an unlikely crowd.

Authors of Wikipedia math pages aren’t contributing to this populist movement. They are intent on impressing each other; competing to see who can reduce a mathematical concept to its most accurate, most precise (and least comprehensible) definition.

A debate rages on a “new way” to do subtraction. Oh does it rage. But step back from that debate and consider that these tricks, algorithms, processes, hacks, become less relevant as new tools take their place. When calculators entered into the classroom, something started to change. That change is still underway.

Do students no longer need to learn to do math by hand? No. But calculators (and computers) have changed the landscape.

Rogue amateur mathematicians, computer artists, DIY makers, and generative music composers are creating beautiful works of mathematical expression at a high rate – and sharing them at an even higher rate. This is a characteristic trait of the “new power“.

Technology

(1) Computers are better at number-crunching than we are. If used appropriately, they can allow us to apply our wonderfully-creative human minds to significant pattern-finding and problems that we are well-suited to solve.

(2) Computer animation, generative music, data visualization, and other digitally-enhanced tools of creativity and analysis are becoming more accessible and powerful – they are helping people create mathematically-oriented experiences that not only delight the senses, but express deep mathematical concepts. And they also help us do work.

(3) The internet is enabling a new generation of talented people (amateurs and professionals) to exchange mathematical ideas, discoveries, and explanations at a rate that could never be achieved via the ponderous machinations of university funding, publishing, and teaching. There will never be another Euler. Mathematical ideas now spread through thousands of minds and percolate within hours. It is becoming increasingly difficult to trace the origins of an idea. Is this good or bad? I don’t know. It’s the new reality.

Five things You Need to Know About the Future of Math

According to Jordan Shapiro:

1. Math education is stuck in the 19th Century.
2. Yesterday’s math class won’t prepare you for tomorrow’s jobs.
3. Numbers and variables are NOT the foundation of math.
4. We can cross the Symbol Barrier.
5. We need to know math’s limitations.

We can (and will – and should) debate how math should be taught. Whether the “symbol barrier” is a actually a barrier, and whether memorizing the multiplication tables is necessary, no one can ignore the seismic changes that are rumbling underfoot.

-Jeffrey

Riffing Deacon, Dehaene, Rotman, and Tegmark: More on the Nature of Mathematics

Is mathematics an invention of the human mind, or is it a universal “eternal truth” that we humans discover?

I once wrote a blog post about Brian Rotman’s book, Ad Infinitum, in which he advocates “taking God out of mathematics and putting the body back in”.

Since writing that blog post, I have thought a lot more about the reality of math. And one reason is that I have since written a book about a set of fractal curves that I “discovered”. Or, did I “design” them? (read on to find the answer :)

Brain Rotman recently wrote a review he wrote for the Guardian on the book Our Mathematical Universe, by Max Tegmark.

Rotman’s review is quite critical. Tegmark claims that mathematics is the very foundation of the structure of the universe. He goes as far as to say that “Our reality isn’t just described by mathematics – it is mathematics”.

Rotman would be one of a growing number of mathematician/philosophers who take issue with this kind of claim. They would counter that mathematics is a human invention. My first encounter of this idea was through Lakoff an Nunez in their book, Where Mathematics Comes From – How the Embodied Mind Brings Mathematics into Being. This book explores mathematics from the point of view of cognition and linguistics.

All very well and good. Math comes from human brains. But the more I engage in mathematical activities, the more sympathetic I become with mathematicians who believe that they are “uncovering” great truths rather than creating ideas out of thin air. While exploring mathematical ideas, the feeling that I am discovering something outside of myself is very strong. What is the reason for this feeling? What is happening in my brain that makes this feeling so strong?

Why I am Becoming a Mathematical Agnostic

I am just about finished reading The Number Sense by Stanislas Dehaene, who is working on the cutting edge of brain imaging and identifying cerebral structures associated with abstract number and calculation. 

Dehaene believes that math is a human invention.

However, he acknowledges the long reality that precedes us. According to Dehaene, the notion of number is perhaps the most primal mathematical construct that our brains perceive, and we share a rudimentary number sense with other animals.

No surprise here: the universe – at least on the scale of mammals and birds, has clumps of matter and events that we perceive and that are meaningful to us. And of course the biosphere is doing its part to continue clumpifying matter and events, including me writing this blog post.

“Number” could be seen as an emergent property of the biosphere. Animals have evolved behaviors associated with number (as well as symmetry, and other attributes associated with math). In turn they impose their emergent representations back onto the environment as part of an ongoing feedback loop of complexification. Humans have simply taken this feedback loop to a conscious level.

A few years back I read Incomplete Nature, How Mind Emerged from Matter by Terrence Deacon. Although his writing is painful, the ideas in the book are very thought-provoking. What I learned from Deacon makes me pause before heading straight to mathematical atheism. I recommend this book. Deacon points out the many levels of emergence that forged the structures of our universe, and ultimately, our minds.

My conclusion: to say that math is strictly a human invention is taking it too far. Math is not arbitrary. We are made out of structured matter that exists throughout the universe. Our bodies, brains, and minds are tuned-in to that structure. We evolved with it and in it. Our language, as well as our genes, are “about” the biosphere, which is “about” the universe.

On Discovering vs. Designing

In Brainfilling Curves, I chide Mandelbrot for saying he “designed” certain fractal curves. I can’t say that I blame him, considering how clever the snowflake sweep is, for instance. But I prefer to say he discovered it.

And I claim that many of the fractal curves in my book were discovered by me. But it gets a little fuzzy at times. Geometrical objects range in primality from the circle (which no one would claim to have designed) to rare, high-order self-avoiding space-filling curves, many of which I have spent years uncovering. The more rare, the more unlikely the geometrical object, and the more information required to describe it, the more like design it becomes. I see it as a continuum. At the far extreme of this continuum would be a painting by Kandinsky.

I believe that we feel the sensation of mathematical discovery because of the evolution of our brains, language, and culture, from which we cannot escape.

The evolution of our brains, language, and culture are a continuation of the evolution of the structure of the universe. Thus math, while it may not originate from the universe, is a language we invented which is finely tuned to it.

And since we are made of the stuff that our math describes, our math feels…

perfect.