IT’S midnight. I’m still at my laptop, trying to answer readers’ questions about the idea I first pitched a few weeks ago-that seriously large numbers may characterise this century’s science. Questions like, “What difference does it make to people’s lives?”
Struggling with ridiculously large sets of permutations and combinations will probably change the way you take decisions on a lot of things. Coincidences, risk, odds-but the everyday examples all look mathematically woolly. There’s got to be a nice, crisp example lurking among all the possible things I can think. Ah! And how many things might that be? Fetch a large envelope and some hot coffee.
Our brains contain about 10,000,000,000 neurons. But numbers with ten zeros are boring to type and hard to read, so write 1010. Each neuron sends out tendrils, called axons, to connect to about 103 others. The formation of memories, associations and thoughts has something to do with the making of those connections. So I can estimate the number of states a brain can be in from the number of ways all those connections could be arranged, without worrying too much about the details.
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What I know of brain function says it would be silly to treat the whole thing as one unit. Say it works as a number of “modules”-103 comes to mind. Each neuron makes 103 different connections to any of the 107 others in the same module, so it has 107 raised to the power 103 (which is 107000) ways of making all its connections.
Suppose I estimate the number of ways each “module” can be “wired up” as 107000 ways of connecting up each neuron, for a total of 107 neurons. That’s 107000 to the power 107, which I’ll have to write 107×1010).
I don’t know about you, but more than one of my neurons hurts already. To keep the numbers down, I’ll assume the 103 modules are independent-in other words evolution, rather than learning or memory, determines the “trunk cabling” between modules. So my brain and its memories might have grown up with any of 107×1013 possible “wiring diagrams”.
But thinking this thought instead of thinking, say, “giraffe”, probably has a lot to do with the momentary strengths, or “weights” of those connections. Approximate these using numbers between 0 and 9. Mercifully skipping the sums, we conclude that each of those notional wiring diagrams is at any moment in one of 101013 states.
Now multiply diagrams by states, and voila! My grand total of thinkable things and haveable experiences: 108×1013. Roughly.
With what could we possibly compare this myriad? How about the entire Universe?
Consider “quantised” space-time. The “Planck time” is the shortest time span that makes any sense in quantum physics, and it’s 10-43 seconds. The Planck length is defined likewise and is about 10-35 metres. Say the Universe is 1010 years old: that’s 3×1060 quanta of time and an observable volume of about 3.6×10183 “cubic Planck lengths”. So our four-dimensional observable Universe has contained, so far, 10244 “quantum there-and-thens”. That puts a rough upper limit on the number of things that have happened: 10102-and-a-bit.
Isn’t it interesting that the number of thinkable things is absurdly larger than the number of events to date?
It could make you think differently about cultural variety and your place in the Universe-or its place in your mind.
