Kevin asked:

Give an individual the ability to understand the biological outcomes of humanity to an undeniable level
of determinism. A Mathematician, for example my friend Marvin Minsky. Could he determine the
outcome of himself? For example, could a Mathematician calculate the outcome of his own math
calculation before, without calculating it? He could not use anything to symbolize the problem
because he would be doing it. Does it have any significance? I was just wondering. I have not had too
much time to think it out. Seems like common sense, that he could not answer the question without
somehow symbolizing, and actually doing it.

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Wow! You could spend a day just figuring out all the angles to this question. There are four issues
that I would like to deal with first, just to get them out of the way. Then we can get to the core of the
question, and my answer.

1. We are considering a hypothetical subject capable of knowing its own internal physical state to the
   point of being able to make predictions about its future states with hundred per cent accuracy. In
   order to do this, it needs to be capable of knowing its internal state without any margin for error, and
   without omitting any detail. If we allow it to omit a single detail, or allow any margin for error, then the
   predictions it makes will involve cumulative errors resulting in greater and greater inaccuracy as time
   goes on. (This is related to the mathematician Lorenz's famous speculation about the 'butterfly
   effect'.) The point is a familiar one from discussions of free will and determinism. In principle, no
   measurement of physical reality can be error free. So even if my actions are determined, there is no
   way they could be reliably predicted by anyone on the basis of knowledge of my physical state.

2. But let us suppose that our subject does know its own internal state totally, and with hundred per
   cent accuracy, and not worry about how it knows this. Immediately, one runs into the following puzzle
   concerning self-knowledge. A being that knows everything there is to know about its internal state
   must seemingly represent that state of knowledge in some symbolic form. Let us call the
   representation, R. Representation R must be as complex — in terms of Wittgenstein's Tractatusit
   must have the same 'logical multiplicity' — as the state which it represents. But now we run into an
   infinite regress. For now we need a second state, R*, which has sufficient logical multiplicity to include
   both the representation R and the state of affairs which R represents. By similar reasoning, it follows
   that we need a third state, R**, which includes R* and R, and so on.

As an illustration of this, imagine a room with a fireplace, and a picture on the fireplace. The picture is
a perfectly accurate picture of the room, with no details missed out. However, included in that picture
is the picture on the fireplace! If we examine the picture within the picture, then it too will contain a
picture, and so on. This potentially infinite series is plainly inconsistent with what we know about
physical reality. There is a physical limit to how 'fine grained' a physical representation can be. Even if
physical reality did not impose this limit, however, the construction of such an image would set an
infinite task.

Does this show that the idea of perfect self-knowledge involves a vicious regress? The regress is only
vicious if the task has to be undertaken one stage at a time. There is no logical inconsistency about
the idea of an infinitely fine grained representation such as I have described. A further point to make
is that it is not even clear that self-knowledge would necessarily involve this hierarchical structure of
representations (although I have no idea what else it could be).

3. So let us take that on board too. It is arguable that the being with total knowledge of its internal
   state can only exist in a possible world where there is no limit to how fine grained a representation
   can be. Now we encounter the third problem. How can a subject doanything, perform any action, if it
   is capable of predicting in advance every decision that it makes? This is the problem which prompts
   Thomas Nagel in The View from Nowhereto talk of a 'penumbra of ignorance', a necessary blind spot
   concerning the cause and effect process that leads an agent to make each decision that they make,
   as a presupposition of our having 'free will', of our having a conception of ourselves as agents.

One reason I am not totally convinced by this is that the physical prediction will not be given in terms
that one would recognize in terms of the language of human action. For example, the action which I
would describe as, "I will go down to the corner store to buy a six pack of beer" will be predicted as a
complex series of bodily movements. It would be perfectly possible to know what those bodily
movements were going to be, without grasping their upshot. I would not recognize myself, or my
actions, from a purely physical description.

4. The last point to clear out of the way specifically concerns the idea that one might calculatethe
   outcome of the physical process which embodies the action of calculating the solution to a maths
   problem. This calculation too involves a physical process, which would itself be capable of being
   calculated. Is there a vicious regress here? Not necessarily. One might imagine an individual whose
   brain was partitionedin such a way that the first partition did the actual maths calculation, the second
   partition calculated the outcome of the physical process taking place in the first partition, while the
   third partition calculated the outcome of the physical process taking place in the second partition, and
   so on.

Incidentally, the description of the physical process involved in calculation will not require any
reference to 'symbols', as you seem to suggest. Numbers appear simply as marks on paper, or
shapes represented in the brain, without any associated meaning.

Now we get to the core problem.

Our subject's brain is partitioned, as described in 4. Presented with the algebraic equation, "x2 - x3 +
x = 49". The subject calculates that the outcome of the brain process involved in solving this equation
will be that the subject writes on the piece of paper, "- 3.634". (This result is incorrect, but I wouldn't
bother trying to find the correct solution!) So now the subject knows the outcome of the calculation, or
do they?

To know the outcome of a calculation you are going to make is not the same as performingthat
calculation, for the following reason. It is possible that in attempting to make the calculation one will
make an error.Performing a calculation involves following certain rules, while predicting the outcome
of the physical process involved in calculation involves different rules. Trying to get a calculation right
is a different task from trying to make a correct prediction about the answer, perhaps the wrong
answer, that one will give.

In terms of our scenario, this result is mildly paradoxical. If I try to predict the outcome of the
calculating process I intend to perform, it seems I have given up on trying to get the calculation right.
Yet, by hypothesis, it is not going to make the slightest difference if I do try to get the calculation right,
because the result will necessarily be the result which I would have predicted.

Geoffrey Klempner