Assumption 1: Minds exist.
This is not an assumption that can be proven. It is far too fundamental for that. Without this assumption, it is impossible to prove anything about anything. However, there is certainly a great deal of circumstantial evidence in support of the existence of the mind, and no better argument than the one provided by Descartes: "I think, therefore I am."
Assumption 2: Minds can be described as a mathematical structure.
Compared to the first assumption, this one is far more radical, and it is easy to believe that it is not true. Need our minds be entirely contained within our brains? If so, can our brains even be described mathematically? We presently do not have a model of the universe which properly describes the interactions of our fundamental particles without inconsistencies. It is in fact entirely possible that we do not live in a mathematical universe.
Nonetheless, the examination of mathematical minds is of critical importance in this century. We are likely on the cusp of discovering true artificial intelligence. Even if our minds are not mathematical in nature, any mind that is implemented within a computerised system will be. I therefore wish to discuss what a mathematical mind implies, and what this means for the nature of our universe.
Any mathematical structure or entity is symbolic in nature. Symbols are difficult things to grasp. When you write a number on a piece of paper, the piece of paper does not contain that number; it merely holds a representation of that number. The number itself is an abstract concept, separate from any part of reality. Nowhere in this universe will you find the actual number 1.
However, just because something is symbolic doesn't mean it can't impact reality. The screen on which you are reading these words is receiving symbols in the form of electrical impulses, and sending symbols in the form of patterns of light and darkness. In turn, you are looking at these symbols with your eyes, which translate what you see into more symbols, sent as electrical pulses down your optic nerve. Everything you see is a symbol interpreted by your brain, and the same applies to everything you can see, touch, or smell. If the nerves are severed, no symbols can reach you, and you will never see anything. In computing, we call these symbols "inputs".
Your muscles are controlled by symbols as well. Everything you do in life is triggered by electrical impulses sent from your brain, each one a tiny symbol. "Contract." "Expand." "Contract." Similarly, these symbols are called "outputs".
The important thing to know about symbols is that the things they represent are not a part of the things that constitute them. A number written in ink is not contained within the ink. If the ink is washed away, the number still exists. It's simply not written with that ink anymore.
A mathematical mind is, in principle, no different. To be sure, if a mind can be represented mathematically, its structure would be fiendishly complex, with billions of connections and thousands of inputs and outputs. But we know that each of those inputs and outputs can be represented symbolically, because they already are. And the size of a structure does not change its fundamental nature. As long as we accept the assumption that the mind is mathematical, we must conclude that it too is symbolic.
You may now be asking: How can a mind be a symbol, when symbols are unchanging? The number 1 is always the number 1, regardless of how or where it is written. Adding another number to 1 produces a third number, but it does not change what 1 is. And all of this is true. In order to represent a mind symbolically, we must view the mind-symbol as a snapshot of that mind in time. This snapshot would contain all of the mind's present thoughts, memories, instincts and desires. And because this snapshot is a mathematical structure, mathematical operations can be performed on it to deduce what the mind would do next, if provided with some hypothetical input. These operations would produce a new snapshot of that mind, much as adding two numbers produces a third. This new snapshot is an entirely separate structure, completely independent from its predecessor. Only the memories that this new snapshot contains provide a link to its past self.
While perceiving a mind in this fashion may be difficult to accept, it's not grossly out of proportion with generally accepted truths. For example, you are not the same person you were seven years ago. In that time, you have learned new things, encountered new experiences, and if nothing else, been impacted by the events of the world around you. The same things happen on smaller scales: while you are largely the same person you were yesterday, there will always be subtle differences. Viewing oneself as a succession of minutely different yet totally separate symbols, each occupying only the slightest moment in time, is merely this process taken to its logical conclusion.
We now have a truly bizarre sense of what a mathematical mind must be: a series of unchanging complex symbols, each one individually inert, yet by a simple (though admittedly long) process of mathematical manipulation, consecutively linked together into a living, thinking, and perceiving whole. It is entirely possible that this is the nature of our own minds—and if they are indeed mathematical, it must be—and at the very least it will certainly be the nature of any artificial minds that we build. And so, the question must be asked: how can such a structure possibly perceive anything? Certainly, mathematical operations can be used to provide input to the structure, and further operations can be used to deduce how that input would affect its memories and actions. But what, precisely, is experiencing these things? What links it all together? Where is the soul?
This question is tricky largely because of how it dodges strict definition. A soul is often described as "that extra something": the piece that cannot be accounted for when all the measurable parts of a person are quantified. When a person is described as "greater than the sum of their parts", the soul is what makes up the difference. We are tempted to believe that without some ineffable extra piece, a system made of simple parts cannot exceed the complexity of its most complex element, but just as with ink molecules forming the number 1, it is the arrangement of those elements which provides its complexity. Even when we view a mind as a sequence of structures, each of those structures is itself staggeringly complex, with innumerable connections, redundancies, and feedback loops between every cortex. Certainly it is not beyond the bounds of reason to imagine such a complex system being capable of processing sensory input into coherent experiences. And if a mathematical mind is capable of perception, this must be precisely what is occurring.
What would it mean for a mind to be nothing but symbols, for its actions to be deduced by mathematical operations? For one, this would mean that determining the actions of such a mind would require nothing more than the solving of equations. It is important to recognize that solving an equation does not create anything new. Finding the solution to an equation is an act of discovery, but the solution was not nonexistent prior to its discovery; rather, the solution is intrinsic to the nature of the equation. Given the exact description of a mind at a certain point in time, provided with a given set of sensory inputs, an equation will result whose solution is foregone, unchanging, and absolute.
Indeed, we can imagine this happening for any number of possible sensory inputs. How would this mind react if gravity suddenly reversed? If red became blue and blue became yellow? If the love of its life suddenly proclaimed their love for it? As long as one has the exact structure of a mind, one can discover the answer to all of these hypothetical scenarios, and therefore, the answer to each of these hypothetical equations would already exist.
Mathematical minds are adrift in seas of possibility. They are, each of them, self-perceiving structures whose nature is abstract: not bound to the rules of this universe, but only to the rules of their own inner workings. The answers to every conceivable hypothetical scenario that could befall them already exist, regardless of whether they have been discovered. And each of those answers results in its own minutely different yet completely separate self-perceiving mind.
To be continued...