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.

————————

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.

Advertisements

The feeling of consciousness is an illusion

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

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

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

Qualia

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

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

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

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

An evolved illusion

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

It’s all an illusion.

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

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

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