The Space of Innovation Possibilities: Pn=Na(Na -1)/2.

‘Infinite possibilities, in infinite combinations.’

Vulcan proverb.

Global economic trends present immense open fields of verity and opportunity for billion dollar commercial markets. On the one hand the developed west has great promise and scope over the coming decades for pioneering ventures and solid growth economies; while the emerging southern hemisphere and far-east offers a rapidly growing spectrum of supply and demand for goods.

But an underlying question remains, ‘How far can all this go?’ ‘How large can the world economy become?’ ‘How fast can technology evolve?’ ‘How much innovation can we achieve before we run out of ideas?’ ‘Are there limits?’

To begin to answer these questions we need to explore the underlying mechanics of innovation and growth.

The Secret of Synergy and Exponential Growth.

I say covert, because very few business books on innovation describe the underlying principles that give such stupendous magnitude and growth. And even the books I have come across that do tell of the embedded codes, stop just at the point where it get interesting. Because I can assure you, there is vastly more to come both economically and technologically than anyone can imagine.

Waves of fundamentally new scientific knowledge, scientific method, and technological breakthroughs are emerging, improving and diverging faster-and-faster in  price-performance ratios. This fact is now a known, and has become part of the standard technological worldview. This in itself is truly a wonderful occurrence and insight.

But again, there is even deeper to understand! Because as price-performance improves, we can observe that new possibilities open up, where in turn the potential for new opportunities expands dramatically.

Simply put, it all boils down a phenomena that gives rise to Synergy: the occurrence where the whole or outcome is larger or greater than the sum of the individual elements.

Synergy comes about as the by-product of growing, interconnecting networks. Which of course we are experiencing today in every quarter of our lives. This is driven by the increase in the number of nodes in a network. As the number of nodes in a system increases in a linear fashion; so the increase in the number interconnection possibilities goes up in an exponential mode!

Here is an equation that begins to show the logic behind exponential growth in networks:

Pn=Na(Na -1)/2

Where Pn equals the potential total network and where Na is the number of nodes or agents. As the number of nodes accumulate linearly, the number of interconnection possibilities goes up geometrically. For example, take 4 nodes added in series: 1+1+1+1=4. You get a linear sum of 4.

Only, if you interconnect each node to each other, you begin to see a nonlinear synergy: 4a(4a-1)/2=6. As the number of nodes go up in linear series, a surprising number of interconnections possibilities emerge.

10a(10a-1)/2 = 45 connections possibilities.

100a(100a -1)/2 = 4,950

1000a(1000a-1)/2 = 299,500

10000a(10000a-1)/2 = 49,995,000

100000a(100000a-1)/2 = 4,999,950,000

Interconnect 1 million nodes, and: 1,000,000a(1,000,000a-1)/2 = ~4.99¹¹. One-million nodes would add up to over 499 billion interconnections.

Now, think of this in terms of ideas and the space of innovation possibilities for yet more new ideas. As new, and even mature ideas join a network, so the number of new connection possibilities open up. As these possibilities open up the space potential increases for the growth of yet more new ideas (whether new tools, new materials, new medicines, new products, new processes, new methods, new ways of seeing and understanding, ad infinitum).

And that stands as a model for technological (idea) networks that provides ever higher performance in creating new market value. And that is where productivity and efficiency gains come from as well. Richer emergent synergy, enabling yet more interconnections, geometrically.

The Laws of the Speed of Technological Innovation.

Moore’s Law, Nielson’s Law, Kurzweil’s Accelerating Returns Law, are enabled by the underlying mechanics of the space of innovation possibilites: Pn=Na(Na -1)/2.

To be brief, Moore’s Law; describes the constant that the number of potential transistors doubles every 18 months within the same geometric space until the scale begin to hit the weird world of the quantum space (10^-19 mm).

Nielsen’s Law says that the amounts of packaged bytes that can be sent down a line will double every 18 months.

Ray Kurzweil’s Law of Increasing Returns; says that technological evolution will accelerate by an exponent over time.

Harris’s Law of Increasing Diversity (me); says that technological and biological diversity expands in proportion to the number of potential interconnections.

Yet all of these laws are empowered by Pn=Na(Na -1)/2.

How Far Can Innovation Go?

Question: if there is an exponentially expanding and accelerating space of possibilities for innovation, where is the final wall? Is innovation finite or infinite?

Let us play a game – yes you will feel silly. I want you to say out loud the word ‘trillion.’ Okay, that was easy. Now hold up both of your hands and on each finger count and say the word ‘trillion’ thirty times. I will do it with you. Okay, hold up your hands in front of you, and off you go, and in a nice singing pitch voice:

‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’ – ‘trillion’

Okay, that was both easy and very silly!.... But no in fact, it was not!!  Because according to a very large number of distinguished scientist’s interpretation of the conventional laws of astrophysics, that is the number of seconds until all the Stars in the universe burn out. Otherwise known as the ‘heat death of the universe,’ where the contents of the cosmos cools down into a very big - so very, very big that it make your head hurt - nippy miasma! Eat your heart out Hitch Hikers Guide to the Galaxy.

To be more precise, since the spatial and temporal resolution of the universe is limited to the very small Planck constant, the number of interconnected solutions is not literally infinite but merely vast.

So, over deep time more innovation will happen beyond any number we can imagine right now. As I say in a post below, life extension technology is coming. What great, amazing, ore-inspiring innovations are we going to experience? The mind boggles!!!