The Mind as a Mathematical Function: How our perceptions are image representations of what truly is and how we can never have any knowledge of the thing in itself.
Hang with me, this is going to be great!
In mathematics, a function is defined as a mapping from one set of elements, to another, where every element of the first set is mapped to a single distinct element of the second set.
Here is a little figure to visually represent what a function is. (For now, forget what a bijection is)
Above in Fig. 1 we have two ‘sets’, as depicted by the circles, and each set contains ‘elements’ (i.e. things). The first set contains five elements, each being a distinct shape. The second set contains five numbers, each being distinct. The figure satisfies the definition of a function because EVERY element (shape) from the first set is mapped (i.e. “used”), and there does NOT exist an element in the first set (i.e. a shape) that is mapped (as depicted by the arrow) to more than one element in the second set (i.e. there doesn’t exist any shape that has two or more arrows coming from it).
So, again, a mathematical function is a mapping of ALL the elements from one set, to the elements of a second set, where only one distinct mapping occurs for each element from the first set.
In Figure 2 we see that this is NOT a function. Do you see why? It is because one of the elements from the first set (‘0’ from input) is mapped to more than one element in the second set(‘-2’ and ‘1’ in the output).
Mathematics, and any field of study, is filled with jargon, language and definitions. I would like to provide some terminology before continuing. The “first set” from which the function does the mapping, is called the “domain”. The elements within the domain are called the “pre-images”. The second set, to which the function does the mapping to, is called the “co-domain”, and the elements within it are called the “images”.
Using our new found vocabulary, we can see from Figure 1 that the domain of the function is a set of shapes, and the co-domain is a set of numbers. We can also say that, for example, the image of the star, with respect to the function, is the number 1. As well, we can say that the pre-image of the number 1 is the star shape (with respect to the function).
Now, most of us, when we think of functions in terms of math, think of something like this:
Figure 3 depicts a graph of a function y= x (squared). The domain is the set of real numbers, and the co-domain is also the set of real numbers. For any given element in the domain (i.e. any given pre-image), there exists one and only one element in the co-domain (i.e. image) that it is mapped to. An example would be mapping x = 0, to y = 0 (since y = 0 (squared), and 0 squared = 0, therefore y = 0). So in this function, 0 maps to 0, and 1 maps to 1, and 2 maps to 4, etc, etc, as depicted by the graph.
Now, when we draw graphs and figures, as presented above, we can see what elements exist in the domain (the first set) and what elements exist in the co-domain (the second set). We see the pre-images and the images. But in real life not all functions work like this.
A function that we can DEFINE is perhaps not always common in real life. Figure 4 shows another function. Here, we have inputs, and the function (“function machine”) acts like a black box, and just gives us the outputs, though we may not know how the function did so. Perhaps the person that engineered the function machine knows (if such a person exists), but we often do not.
We surround ourselves with machines that are functions. For example, a tv remote. One function on the remote is the volume up button. If I press the volume up button, it maps to a distinct event, that being the audio output of the tv increases by a single interval. If, for example, sometimes when I clicked on the volume up button the volume went up by 5 intervals, or down by one, or it caused the tv to change channels, or all together, this would not be a function, as the single input (pre-image/element “up button press”) would map to different things (image/element tv channel change, etc, etc). Now, for most of the population, the tv remote is a black box function. We don’t know how this mapping works, we only know that it does it. But somewhere, someone out there does know.
In both the mathematical graph examples and the tv remote examples we were aware and had knowledge of both the inputs (pre-images) and the outputs (images) of the functions. Is there not instances of when we are ONLY aware of the outputs (images) of a given function?
Let’s say I had a graph, or a table containing the coordinates of a graph available to me. And suppose I was with someone who didn’t have access to such a graph or table. Perhaps I am with you, and you are that person. Suppose I were to read to you all the images of a given function, such as the one from Figure 3. Suppose I skipped out the decimal numbers and only listed the integers (whole numbers) starting from 0, that were images (y values) of the function y = x (squared). Suppose I read them in increasing order. I would say “zero, one, four, nine, sixteen, twenty five…”. Now, reading them in order, and only having information of the images of the function you MAY be able to deduce what the actual function is, and in doing so, what the pre-images might be. “AHA! Each number you mention is the square of a number… so the pre-images are zero, one, two, three, four, …. “.
But what about the function from the Figure 1? Listing to you the images “one, two, three, four, five”, you might infer any number of functions, and thus any number of pre-images. It would be highly doubtful, though possible, that you may be correct in hypothesizing that the pre-images were distinct geometric shapes. But that would simply be a lucky guess, and not an expression of true knowledge.
Likewise, what if I were to return to my original function from Fig. 3, and instead listed the images in random order, “one hundred forty-four, four, eighty one, ten thousand…” etc? It would be possible, though not likely, to correctly state what the pre-images were, and what the function was. For all you know the function is simply a mapping of any number to itself (i.e. y = x), or again, from shapes to numbers, or people to numbers (as we do for identification cards), etc.
Now, this is very important. When we have only perceptual knowledge of the images of a function, it is impossible to infer any knowledge as to the pre-images (the input) to the function. The images are perceived, and the function remains a black box, with the inputs, the pre-images, unknowable. We are left with only the inference that whatever it is that they are, they do exist. Nothing more can be known about them.
The Mind as a Function
I am going to make an assumption. That the universe/reality is not solipsistic in nature. Solipsism is the belief that only YOU (i.e. me) exist, and nothing else is real. There exists only one single mind, and everything else only exists inside that mind.
Let’s assume that the universe does not follow that structure, and that there actually are things that populate the universe other than myself. Mathematically speaking, the number of elements in the input set, the domain (the number of pre-images) is greater than one.
Now that you also exist in this universe, I am capable of using the word ‘we’. Since we are conscious beings, we are conscious of something. To be conscious is to be conscious of something. In this moment I am conscious of my typing, my laptop, I have a slight need to pee beginning to build, I can hear some buzzing from the walls, and I am conscious that I am about to end this sentence. The objects that we are conscious of include our perceptions (vision, smells, tastes, sounds, tactile sensations) and our thoughts.
I am led to believe that the mind is a function. If, in fact, the mind is a function, then the objects of our consciousness, the things we are aware of, all the things that you can possibly experience, are images that are mapped onto (the outputs), via the mind, from a given input (pre-image).
First, I will have to argue that there exists a distinct mapping from every pre-image to each image. Secondly, that means that whatever exists in the domain (i.e. the universe) is “mappable”.
I believe that it is intrinsic in our existing in the world that we operate with the logical structure that every pre-image produces a single distinct image. This, I believe, is nothing more than the mathematical description of causality. Cause and effect. I argue that to deny this is equivalent to an admission of insanity.
There are many definitions to insanity, but a popular colloquial definition is ‘someone who repeats the same process, expecting a different result.’ Is this not nothing more than to deny causality? Implicit in this definition is an individual in the world, acting in one way, and expecting a different effect to occur. We call such a person insane. We implicitly recognize that in order to effect a change, a different causal force must be applied to a given circumstance. In order for a unique effect to happen, one that differs from when we apply input x, we must apply a different input (i.e. try anything BUT x) if we are to have any hopes of observing a new unique effect to occur.
To operate in the world is to implicitly, if not explicitly, be ruled by causality. When we experience hunger we are moved to eat. This is causality. Not only this, the mere idea that IF we eat THEN the hunger will be taken care of, is a logical inference that abides by causality. Why else would anyone ACT in the world, if they did not implicitly believe that the act would effect a change? When someone is upset, we ask “what happened?” Why? Because implicitly we operate under the structure of causality, we make the inference “Person x wasn’t upset before. Person x is upset now. An event must have happened to cause person x to being upset.” Causality.
Since we operate under the structure of causality, it is normal for us to infer that when we “see” or “hear” or “smell” or perceive something, that the perception itself was caused by something. That when I see a book on the table next to me, that the event, the phenomenological experience of perceiving a visual of a “book”, is caused because there ACTUALLY IS a book on the table, or at a minimum, that there is something that inflicts onto me a perception of the book. The perception of the book is a representation of something that, external to me, has an ACTUAL objective existence. That is to say, the perception of the book is an image, and what that image (perception) represents is an actual book (the pre-image).
So, we have the pre-image, the thing that has external existence from myself. That being an objectively existing object in the universe, a book. The mind, a function, maps that book and produces an image (quite an appropriate term, I would add) that is my perceptual phenomenological experience.
Causality is important because, when I turn around and return back again to the book, the book still looks the same. It hasn’t changed. It isn’t a book one moment, and then a tissue box another. It isn’t a book one moment, and a blanket the next. If the pre-image of what is mapped, via my mind, to “the book” were to change to something else, I would have evidence that the mapping is not to one distinct image, and so the mind would not be a function. Also, it is a book and only a book. It is not both a book and a basket. It is simply a book. Because I operate under causality, I infer that the thing from which my vision of “the book” emanates from (i.e. the physical matter), is a constant thing, and it is a single thing, and thus, it maps to only one image. Because my phenomenological experience is consistent, I can only infer that the mapping is distinct. With that, the things we perceive are images of a mapping, via the mind (a function), from objects that exist in the world, external to us.
To argue that all things in the universe are mappable, I will save for another day. It is sufficient for my purposes here in this post to deal with the distinct mappings of the mind from objects/pre-images in the universe to perceptual phenomena/images.
Can we know about the “objective” universe?
Remember that function y = x (squared)? Remember the thought experiment of only being aware of the images, and not being able to deduce, to know, what the function was? What the pre-images were? What the objects were which caused the existence of the images, via the function?
Most humans believe in objective truth and that objects in the universe have an objective existence. That if all humans were removed from the universe this very instant, the book on the table would still exist, as I had always perceived it. I ask myself the question: What does such an assertion even mean?
Our perceptions are the only things we are capable of experiencing. Every sight, every smell and every thought is a perception that “exists” in the mind. If the mind is a function, and our perceptions are images that are mapped from external inputs, then, again, I ask: What does it mean to say that ‘if all humans were to cease to exist that the book on the table will still exist as it does now’? This is the same as asking what does it mean to say that ‘if all functions were to cease to exist, the images of the functions would still exist’? My answer: it means nothing. The question is a meaningless one.
We perceive the world through the mind. The world, the universe, is mapped to our perceptions via a function, the mind. We ONLY ever experience and have knowledge of the output, of the images of the function. To make the inference, the belief, that what we perceive is actually what exists is to say that the images of the function are equal to the pre-images. Meaning, the function maps every element to itself (i.e. y = x). “Of course the book that you see REALLY IS the book! Of course it is!” That would be the reply of most, I believe. But we have nothing but empirical evidence to the contrary.
We have so much diversity in perception among living organisms. Some species can perceive greater or lesser ranges of the electromagnetic spectrum (black and white vs. color vs. ultraviolet), of sound frequencies, of scent. We have color blindness. We have synesthesia. We have people who taste colour, and people who see sound. We have people that can see objects, like the book on my table, except those objects are numbers. When asked to compute 239487239847239 * 293472938472983 + 2347 divided by 12322, there are people, human beings, that see structures, objects, representative of those numbers in front of them, and they don’t do ‘math’ to calculate the answer, they just look at the resulting structure and see it, the way I see the book on the table.
With such empirical evidence for diversity of perception, who (or what organism) can say which one is “objectively true”?! It is meaningless. The mere fact that perceptions can be different simply shows that different living things capable of perception have different functions. The images of those functions are different because the functions (minds) are different. To say any one set of images (perceptions) are the objective truth is nothing more than bias and sheer egoism and adoption of convention.
What does have “objective” existence are the pre-images. The set of ‘things’, external to us in the universe, that the mind maps to our perceptions. The thing is, we cannot have any knowledge of them, as they actually are. It is a pure impossibility. We can only come to know them through the mind, through a function, through mappings to our perceptions, through images.
If tomorrow all conscious beings were removed from the universe the book would not continue to exist. What would exist, would be something, and that something would be the pre-image that maps to what I call the book. Any meaningful description of that pre-image is unknowable, other than the inference I make of its necessary existence.