IS IT just me, or are numbers getting bigger?
Cast your mind back to the 19th century, virtually speaking. Consider a few
of the most important numbers in the science of the time. Thermodynamics
developed around numbers such as the critical point of water鈥攁t which
liquid and vapour are indistinguishable鈥攏ear 647 kelvin and 218
atmospheres. Humans have 23 pairs of chromosomes. And the founders of
electromagnetic theory simply defined their key number鈥攖he 鈥減ermeability
of free space鈥濃�攖o be 1.
I鈥檓 cheating a bit, of course, leaving out, for example, Anders 脜ngstr枚m
and his unit of length. But look at some quintessential 20th-century numbers.
The speed of light is 300,000,000 metres per second鈥攎ore easily written as
3 脳108. Don鈥檛 write the zeros, write the number of zeros.
In the 20th century,
we found that there are about 3 脳 109 base pairs in human DNA,
our Universe is
about 1010 years old, that there are about 1011
stars in our Galaxy. . . and
there are 6 脳 1023 atoms in a gram of hydrogen.
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And this new century looks set to deal with a whole new kind of number.
Molecular biology is going to have to look at questions like the number of ways
that 30,000 genes can interact with each other. First, if you assume that each
can be either on or off, that implies 230,000 possible states of the genome,
about 109000.
Then, to deal with the number of ways you can arrange genes in networks that
affect each other鈥檚 expression, you need to use numbers such as 30,000
factorial: that is, 30,000 脳 29,999 脳 29,998. . .
脳 3 脳 2 脳 1. According to my
calculator that鈥檚 10121,287. And I鈥檓 not going to check by hand that the sum
didn鈥檛 break it.
You could say that 19th-century numbers were typically written N, and
20th-century numbers 10N.
How many people will end up using a new, higher-order
notation for quick calculations on 21st-century numbers? Don鈥檛 write 10100,000
but 10鈭�105, to indicate that there are five zeros in the number of zeros.
This new order of calculations raises new problems of communication between
scientists and non-scientists, and between scientists of different generations.
We already have enough trouble visualising 106, though it should be simple
enough: just think of the number of sugar lumps in a 1-metre cube. Many
arguments about evolution founder on the difficulty of appreciating what might
happen in 105 generations of natural selection,
let alone 1010. Worse, I
notice a difference in world view between people who鈥檝e built and used a
six-dimensional object (on a computer) and those who haven鈥檛.
Multidimensional spaces provide one of the most useful ways of dealing with
interactions among 30,000 genes. Consider an abstract space with 30,000
dimensions: each point is a possible interaction. To consider how a mind may
inhabit a large network of neurons in a brain, start with a space of 1010
dimensions. . .
鈥淢ore is different鈥� is the rallying cry of those who proclaim that evolved
forms or consciousness are 鈥渆mergent properties鈥� of very large systems. Sceptics
riposte that this is arm-waving nonsense.
But what we have here, I suggest, is a very strong hint that in the 21st
century we need a radically new understanding of what we mean by 鈥渕ore鈥�.