Can people, in principle, be modelled mathematically?
by Bruce Dickson
You ask for a cogent reason why it's possible to (at least in principle) understand some complex phenomena yet why it may not be possible to understand people.
I suppose the reason comes down to complexity .
[ 1 ] Humans interact with their surroundings – so do charged particles – so the argument that the probing perturbs the subject is actually specious – it works well enough for quantum mechanics at least in the sense that a statistical measurement can be made.
My view of the sciences is that in explaining a phenomenon, we attempt to build a [principal component analysis] model based on a minimal set of principal components. This is a more exact expression of Occam's razor – the simplest model is probably the best.
I think of principal components as eigenfunctions spanning a space of all that is possible. This suggests that the number of components would be very large if one chose to examine the universe (something like 51080 or so.) If you like, the number of principal components is precisely equal to the number of eigenvalues associated with what you are trying to study.
This works for many physical things because we can make the approximation that the system is isolated – so that for example the phase of the moon does not play much role on my car's petrol consumption. I must stress that this is a deliberate approximation we make just so that the size of the problem becomes tractable.
Mathematically, I think in terms of diagonalising an incredibly sparse matrix, with perhaps half a dozen significant terms and lots of 1×10−20's or whatever. Not zero, but very, very small indeed, so that the significance disappears into the noise of measurement.
When one starts playing with the microscopic, it becomes difficult to determine how one's probe is interacting with the subject. A macroscopic version of this would be "describe the internal structure of a marble by examining them with a broomstick with a blindfold. and listening while Pink Floyd roars at 120 dBA." It's fairly easy to see that we can actually do very few experiments that tell us anything about a specific marble – but with care we can work out the average properties of marbles, and perhaps even some statistical distribution that gives us a measure of confidence in our results.
[ 2 ]: It's possible to conceive of something which cannot actually fit inside the universe – large numbers are a good example. How can you count to a googol (101010) when the universe doesn't have that number of particles.
[ 3 ]: This is intimately related to complexity theory and in particular cryptography which uses state machines to generate cypher streams.
[ 4 ]: Simple example of this – some folk are buying t-shirts claiming that Pluto is a planet. The IAU meeting which was filled with people who they have probably never met has influenced their eigenstate. Even in flights of wild fancy, I would not have put Pluto's planethood on my list of PCA's to describe humans but I don't see what other effect would cause them to print or buy or wear a similar t-shirt.
Enter people. It is estimated that the human has ±1010 synapses. Each synapse can be in two states – conducting or not – so it is arguable that there are ±10102 eigenstates available. This is quite a large number – vastly greater that the number of physical eigenstates in the entire universe . We can perhaps measure the bulk properties of an eigenstate – but it is formally impossible to determine what the probability of encountering a given eigenstate is because we have no way of counting them (the possible number exceeds the number of particles in the universe.)
I'm not suggesting for a second that each eigenstate comprises a unique "feeling" nonetheless it is fairly apparent that PCA is at best difficult because we can only determine the components in a probabilistic sense – we have no formal reason for believing the set we choose to be complete or even reasonably representative because under the incompleteness theorem some eigenfunctions cannot be deduced.
So here is the problem – without a demonstrably sound set of eigenvectors it is basically impossible to understand a sufficiently complex state-machine . We don't have such a robust set, nor can we ever demonstrate in principle that a set is robust . The set of eigenfunctions – or at least their associated eigenvalues – is likely to be different for all people . Humans are dependent of the synapse arrangement – they are therefore state-machines. Ergo humans are difficult to understand, and it may be impossible to prove understanding (or lack thereof) even if you have a model.
nothing more to see. please move along.