There is a shadow now, through the windowpane, of a bending branch of twisted rain. It makes me think of neural nets, the way the rain drips down in rivulets of broken lines, in fractal sets like lightning, branches, or unspooling twine. Through the blinds, in threadlike slits, the silhouette of maple boughs is segmented into shaded rifts, and my room, like Plato’s cave, shows me only slices of reality.

Beyond the curtain, in the shadowland, I see shapes that melt, that meld, that dissipate like silt or sand. The semblance of those forms is mine to to mold and shape; a knight in armor, or a hero’s cape. What shapes stir thereout, in that shadowland! A whirling dress, a regent’s hand. Through that world my princess twirled, her materiality her curse, and my demand.”

-William Arthur Grant

The late, great, and little known William Arthur Grant, doctor of philosophy, and incidentally, my grandfather, took quite a different position than I do on many things. But it’s nonetheless fascinating to go through the papers he left behind him. One set in particular, bound in leather, begins with the quotation above, and unfolds a beautiful and somewhat eerie philosophy in his particular style of writing, not quite either verse or prose. It was given to me when he passed away when I was in seventh grade.

I really don’t know what to make of it, honestly. I’ve had it in my room since I was twelve years old, and have kept it with me wherever I’ve gone, even now, as a young adult living on the coast of Lake Michigan. Sometimes it seems to be a sort of narrative, and other times it’s a sort of essay on various philosophical subjects. Whatever it is, it’s probably what inspired me most to want to be a writer.

I think what I liked most at first, before I could understand its content, was the inky pen he used on the heavy yellowing paper. It was the sort of pen that geniuses always use in old sketches and journals, and he had a fluidly messy, though very orderly handwriting that looked like something adventurous or at least very interesting. I liked the book itself before I liked the book. I only barely tread into its pages far enough to glean any meaning.

As much as I would like to reprint his entire text, I think that it would somehow fail to convey what the book itself is. The fact of the book is more than the story in it, it’s the pages, smudged by his fingers and worn yellow in a dusty attic. The original intent of the physical object that would become the book was that it would be some sort of ledger. There are dotted lines to give structure to the writing and the pages are dated down every quarter. The dates are ignored, sometimes written over completely. I don’t think I could recreate the experience of reading it online.

This is an old but important question in philosophy, and one which I think has a straightforward answer which is generally overlooked. If I were to make copies of the book in computer text, and send it online, would I have spread the book, or a ghost of it? What is the important feature of a text that we label it such? Is “Great Expectations,” or “Huckleberry Finn,” a collection of physical books, or is it some abstract object, which may pop in or out of existence whenever certain books are published? If I were to burn every copy of “Les Miserables,” would that transform “Les Miserables” to dust and ash, or would it only banish it temporally from our realm, ready to be recalled whenever a new author happens to put those same words in that same order?

If we extend that line of thought to people, what is it then, that makes a person a person? If you burned my body, would the potentiality of my body die? If the atoms of my body were reconstructed, would that be a new me? If a computer program was written which exactly had my memories, my thoughts, and my voice, would that be me also?

As I said, I think that the answer is straightforward. In the case of “Les Miserables,” there are a number of distinct physical objects which exist that we call books, and when we look at some of them, we call them “Les Miserables,” and when we look at others, we call them, “Slaughterhouse V.” The relevant issue is whether these books can get a human mind to light up and call them by that particular name. If I read “Les Miserables,” online, what I’m doing is interacting with a certain physical object, in this case a computer, which has been designed to manipulate my mind in a way which evokes, “Les Miserables.” We get caught up in trying to answer questions like, “What is Les Miserables, really,” when the reality of the situation is very plain. There are a large number of separate physical objects which have certain characteristics, and we have certain categories for placing these objects in. Some of the categories have names like “Les Miserables,” and some of these categories have names like, “Huckleberry Finn,” but that doesn’t mean that anything metaphysically significant is happening when one of these objects is born. It’s not a mystery that some of these objects may exist as ink and paper, and some of these objects may exist as computer circuits and pixels on a screen. Those two things are, in actuality, two different objects. We just put them in the same category because they have the same property of being able to get us to think about John Valjean or Billy Pilgrim. “Les Miserables” isn’t an abstract object existing in some other realm, it’s an idea in our heads that we apply to certain objects which our brains interpret as being similar.

In a similar way, some disagreeable (yet very friendly) person once asked me whether destroying a computer running windows would destroy windows, or just the computer. Of course, I would have answered, if I had thought of it at the time, that it wouldn’t destroy “windows,” because windows is an idea about things which exist in our heads. It would destroy one particular object which we interpret as having the quality of “windows,” but “windows” itself is just a category that we have for putting those objects in. The real question is whether destroying every human being on the planet would destroy windows. If we were to go extinct as a species, but leave all of our books and computers behind, would “Les Miserables” still exist? Again, the straightforward answer is that there would still be a bunch of physical objects, and nobody to put them into different categories. Whether one book is “Les Miserables’ or “Huckleberry Finn” at that point doesn’t mean much of anything. They still retain the qualities which would make conscious beings call them one or the other if they were there, but in the absence of conscious beings, those prospects simply no longer apply. It’s a bit like asking whether things could still get wet if there were no liquids. No, they couldn’t. Everything would still have the properties which, when interacting with liquids, creates, “wetness,” but without liquids, “wetness” would never happen. “Les Miserables” exists in the interactions of a mind and an object, and without one half of the recipe, it’s meaningless to talk about whether it would exist.

This brushes against another topic in philosophy, namely, mathematical platonism. According to the Stanford Encyclopedia of Philosophy, it’s the predominant position taken by most working mathematicians. That may or may not be true, but I’ll take Stanford at its word on this. The Encyclopedia goes on to describe mathematical platonism as the position which holds the following to be true:

There are mathematical objects.

Mathematical objects are abstract.

Mathematical objects are independent of intelligent agents and their language, thought, and practices.

You can tell that these are very serious axioms because they’re so bold. Personally, I think that this is some of the most ridiculous philosophy I’ve ever come across, but a lot of people much smarter than myself are taking it seriously, so there’s a good chance that I don’t know what I’m talking about. That being said, I think that there’s a huge contradiction here in the definition of ‘abstract’. By my way of thinking, recorded above, there’s no such thing as an object which is ‘abstract’ and also ‘mind independent.’ To me, it seems like being ‘abstract’ is a way of saying that something falls into a category inside of our heads. Already very suspicious of abstraction, I wasn’t very much comforted by the following definition in “The Internet Encyclopedia of Philosophy,” the self titled peer -reviewed academic resource.

There is no straightforward way of addressing what it is to be an abstract object or structure, because “abstract” is a philosophical term of art. Although its primary uses share something in common—they all contrast abstract items (for example, mathematical entities, propositions, type-individuated linguistic characters, pieces of music, novels, etc.) with concrete, most importantly spatio-temporal, items (for example, electrons, planets, particular copies of novels and performances of pieces of music, etc.)

Any definition that begins with, “there is no straightforward way of addressing what it is to be [the object of this definition]” should be fired from the dictionary. Whether I’m right or wrong, I have at least the conceit that I’m being straightforward. I think this definition is representing a confused thought process more than an insight into ‘abstractness.’ It lists novels and pieces of music as ‘abstract’ entities, and individual books and electrons as ‘concrete.’ Ignoring the fact that electrons hardly exist anyway (a joke, but come on, quantum stuff just seems to come and go as it pleases) I do think that there’s no way to call a novel an abstract object in a way which is ‘mind independent.’ There are a bunch of concrete objects, such as books, and their ‘abstractness’ comes from the mental category that we’re artificially placing them into.

I think the reason that mathematicians are so prone to making what I consider to be a major mistake comes from the foundations of Zermelo Fraenkel Set theory. According to ZFC, numbers are best described in terms of sets, or abstract groups of mathematical elements which have definable relations between them. One of the founding assumptions of ZFC is the concept of the set itself. A “set” is just a defined group of elements, such as the natural numbers. But the concept of the set itself seems to float invisibly, abstractly over ZFC, in particular in two areas. The first is the “empty set,” or the idea of a set which includes no elements. In ZFC, the empty set is equated with zero, and natural numbers are defined as progressing in relation to the empty set. Each progressive natural number is a set which includes all previous naturals, but uniquely, the ’empty set’ is a set which includes nothing. The empty set is the category of observation itself. It is the possibility of existence, the foundation of all ‘abstractness.’ It is the concept of a category of things, but with no things to put in that category, and in ZFC, the empty set must be treated as a real object, because all further sets are defined in relation to it. Therefore, the concept of abstractness is woven foundationally into ZFC, and ZFC would collapse without it.

The second place where I think that ZFC assumes abstractness is in certain definitions of infinity. If there can be a set of objects which is unlimited, which has no end to the number of elements within it, such as is the case with the natural numbers, how exactly do you say that you have a ‘set’ which is a distinct object into itself? Of course, that’s precisely the sort of question which mathematicians seek to answer with concepts such as Dedekind cuts, but I do feel that it tends to beg the question, because all attempts to answer those questions within ZMC must assume that sets themselves are objects. If you’re assuming that sets are real things, and that you can put any little thing that your heart desires into that set, it doesn’t prove much about whether sets exist or not to say, “And also, this particular set has everything in it.” I’ve made a few stabs at addressing this mathematically, but an acquaintance of mine, a lecturer in mathematics at University of Colorado, Boulder, politely tells me that my attempts are so far very naïve. Anyway, I’m not really trying to talk about mathematical platonism here. Let’s get back to minds and books, which I think is better anyway. Stupid math. Wherever platonic objects exist, I hope it’s unpleasant there.

If we keep going with these ideas and apply them to people, we have to eventually come to the question of whether or not a clone of one person with that person’s memories counts as being that person. Every philosopher and science fiction writer has to get there eventually. In real life, we don’t often have to face that question, at least we haven’t had to yet, but there are some areas where it comes up. For instance, is the ‘me’ now the same ‘me’ that existed several seconds ago, or ten years ago? Or is the drunk me who punched you last night (sorry) the same ‘me’ as the sober me who doesn’t remember doing that? Following my reasoning above, I think that the best thing to do is to just calm down about it already, because the reality of the situation is pretty plain and doesn’t need to be as distressing as some philosophers want to make it. Just as we are free to call some books, “Les Miserables” and others, “Huckleberry Finn,” we are free to call some people, “Tom” and others “Tim.” There are no cosmically right answers here, just some answers which are more sensible in some situations and some which work better in others. There’s something distinct about the phenomenon which has occurred, and which I’ve called ‘myself,’ which differentiates it from other similar phenomena, so it’s convenient and sensible to think of myself as belonging to a different category than the one that other people belong in. But it’s still just a made up category, and if I wanted to say that what really defines the character of Michael Trites is that he has brown eyes, then I could go ahead and call everybody with brown eyes by my name. There’s just nothing particularly useful about doing that, and it violates the general public consensus of what makes a person an individual, so I don’t behave that way. If there were other phenomena which were very very similar to me, such as a clone with all of my thoughts, it would seem more justifiable to use the same category to describe both physical objects. Both separate objects, the clone and myself, could say that we belong to the same “set” of Michael Trites, but perhaps that we were slightly different permutations of that same hypothetical set. Or we could both change our names and have nothing to do with each other. It doesn’t ultimately matter, there are no metaphysical constraints here on what we should consider ourselves. There are simply different ways of looking at the situation, some of which are more useful. “Michael Trites” is not some abstract entity which exists by itself. My abstractness exists in my head, and we use that abstractness to refer to, currently, one particular category of physical phenomenon. (This will change once a good storm brews up and powers my equipment.)

Once my clone is alive, however, the question reaches its peak and pique when it come to my death. If I had an active alternate out there, would I be willing to let myself die? Would I live on, in my truest sense, if my clone lived on? Would that be enough? While maintaining that that question is ultimately a matter of interpretation, emotionally, I have to say that I wouldn’t simply let myself die just because some “me” was out there as a clone. I’m attached to my progressing, physical self, a cyclone of matter composed from the substance of reality itself. I don’t think there’s anything indefensible about that. I feel that I’m beyond being more than the sum of my parts. I’m very much emotionally bound to my parts, and that emotional connection is a deep part of the ‘self’ that my every breath and heartbeat have been struggling to maintain all my life. I love my hands, which bear my scars. What would a scar be on a clone? It would be mass of flesh, only incidentally a tribute to a wound. The physical progression of my being is among the most important properties in my definition of who I am. I could abandon that, but I don’t want to.

If I were to make digital copies of my grandfather’s book, I wouldn’t give up the original. The reality of the physical object which is that book is just as important to me as the words that it contains. Yes, if that book were to disappear, I would be glad to have a backup on file, but I wouldn’t feel that I had kept a shade of that book with anything more than a passing rezemblance to the original. There would be no way to retreive that book, no land beyond the curtain I could jump into where I could find it. Once lost, it will be lost, like my best and most honest “I” will one day be.


4 thoughts on “Permutations

  1. For instance, what makes you think that there’s a contradiction between ‘abstractness’ and ‘mind independence?’

  2. Ok, good question. Here’s how I might approach that. First, because people talk about physical objects such as books, and abstract objects such as novels, let’s assume the existence of something called an ‘object.’ Objects might have various properties, and among these properties, we can include abstractness, and concreteness. Concrete objects exist physically. So I could say that a particular apple is ‘concrete,’ as long as we all agree that it exists physically. And then there are other objects, such as numbers, which are ‘abstract,’ meaning that they don’t exist physically. So we have a category of items which are called ‘objects,’ some of which are concrete, and some of which are abstract. Now let’s examine what it is to be an object in the first place. How do we define what it is to be an object? We know that objects can be either concrete or abstract, so we know that you can’t use either of those individually and separately as foundations to describe what it is to be an object. For instance, you can’t say that “Objects are physical items which… etc,” because objects may be either physical or nonphysical. So if we want to know what an object is, we have the daunting task of finding out what sort of category might include things which are both ‘physical’ and ‘abstract.’ One possible category that springs to mind is that objects are all things that we can think about. We can think about books, and we can think about novels. We can think about peter pan, and we can think about an actor who plays peter pan. So if we take that as a definition, then we can further define ‘concrete’ objects and ‘abstract’ objects in this way; Concrete objects are things that we can think about, but which are physical. Abstract objects are things that we can think about, but which are not physical. So far, it’s all fairly straightforward. Now let’s invent a hypothetical division in the category of abstract objects. Abstract objects can be ‘mind dependent,’ or ‘mind independent.’ ‘Mind dependent’ objects are things that we can think about, which are not physical, which only exist because we are thinking about them. ‘Mind independent’ abstract objects are things that we can think about which are not physical but would still exist if we stopped thinking about them. Some items in this category are, supposedly, ghosts, magic spells, and, according to mathematical platonism, numbers. On the face of it, it’s hard to imagine what such entities would be. It seems suspicious to me that there can be abstract objects which are both mind independent and mind dependent, while there’s no such division among concrete objects. There aren’t, that we know of, physical objects which are ‘mind dependent,’ in the sense that they would stop existing if we stopped thinking of them. It seems more likely that ‘mind dependent’ and ‘mind independent’ are actually different titles for, ‘concrete objects’ and ‘abstract objects.’

    Now let’s look at what happens when you change the definition of objects from, “things that we can think about,” to “things that actually exist.” By this definition, concrete objects are, “Things that actually exist and which are physical, such as apples, books, and electrons.” So far so good. But abstract objects are, “things that actually exist but are not physical, such as Peter Pan, Santa Claus, and the number three.” Again, I think it’s suspicious that we keep finding numbers in the same category as ghosts and Santa Claus, and that abstract objects don’t tend to hold up logically when we stipulate that we must be talking about things which actually exist. If we add an addendum to the definition of abstract, so that all abstract objects must be mind dependent, then things start working again. We can say, “Abstract objects are things which actually exist, but only as concepts, such as Santa Claus, Peter Pan, and the number three.” But what we just did was again to match ‘mind dependent’ and ‘mind independent’ with ‘concrete’, and ‘abstract.’

    Let’s try this once more, using the definition of objects which must actually exist, and splitting abstractness into two categories. “Mind dependent abstract objects are things which are not physical and actually exist, but only as concepts, such as Peter Pan and Democracy. Mind independent abstract objects are things which actually exist but are not physical, even when we’re not thinking about them, such as…” and here we trail off, because we don’t know, for sure, of anything in that category. Traditionally this is where ghosts and magic goes. Maybe it’s also where numbers go, too. If this is the correct way of thinking about things, what this means is that there’s a whole new category of things which exist but which are not physical, but it’s clear that whatever this category is, it’s not the same category that Peter Pan and Beethoven’s Fifth falls into. It’s a shadow category, a very mysterious category which I think, as skeptics, we should only believe in in the event of finding great evidence. So far I think that no such evidence exists. The real kicker here is that it’s easy to make this mysterious category disappear if we simply maintain mind-dependence as a fundamental feature of ‘abstractness.’

    Let’s go over the axioms of mathematical Platonism again with this in mind, using the different variations of our previous definitions. First, let’s assume that an object means, “something that we can think about,” and abstract objects can be either mind dependent or mind independent.

    1) There are mathematical objects (We can think about things which we call mathematical objects)
    2) Mathematical objects are abstract (We can think about them but they are not physical)
    3) Mathematical objects are mind independent(We can think about them, they are not physical, they will not go away when we stop thinking about them)

    Now let’s use the definition of ‘objects’ as things that we can think about, and the definition of abstract as being only mind dependent.

    1) There are mathematical objects (We can think about things which we call mathematical objects)
    2) Mathematical objects are abstract (We can think about them but they are not physical and are mind dependent)
    3) Mathematical objects are mind independent (The Universe Explodes)

    As you can see, there is a contradiction here. If ‘objects’ are defined as something that we can think about, then Mathematical Platonism only holds up if you can divide abstract objects into being mind independent or mind dependent. Now let’s see what happens if we assume that objects are defined as being, “things that exist,” and allow mind independence.

    1) There are mathematical objects (Things called ‘mathematical things that exist’ exist)
    2) They are abstract (They exist, but they are not physical)
    3) They are mind independent (They exist, are not physical, and don’t need us to think about them.)

    And lastly, ‘objects’ are things which exist, but no mind-dependent abstractions.

    1) There are mathematical objects (Things called ‘mathematical things that exist’ exist)
    2) They are abstract (They exist, but dependent on the mind, and not physically)
    3) They are mind independent (The Universe Explodes)

    Again, we need mind independence for it to work.

    When objects are defined by their existence, you can only clear up abstract objects such as Peter Pan by saying that they exist ‘in the head’. When objects are defined as being things that we can think about, mind independence gets wiggly. I think there’s something to that. ‘Peter Pan’ falls into the framework only if you add the ‘in our heads’ somewhere into the mix, either at the top, in the definition of ‘objects’ or in the bottom, in the definition of certain types of abstractness. So we can only assume objects to ‘exist’ if by ‘exist’ we allow that they may exist ‘in our heads.’ And if that’s the case, what does it really mean to exist? Let’s examine “existence” more closely. Let’s say that existence might mean, “things that are in the ‘real world,’ not the world of our heads.” So under this definition, saying that peter pan is an abstract object means that “Peter Pan is something which is in the real world and not as a concept but is not physical and is a concept.” Which is an obvious contradiction. That is true if we are talking about the physical correlate to the idea of peter pan which exists in our heads. It is not true if we are talking about Peter Pan in an abstract way. So we can’t define ‘objects’ as things which exist, if we’re determined to define ‘existence’ as meaning things which are actual, and not conceptual, and also determined that “Peter Pan” does not exist. We have to pick one or the other. Either Peter Pan exists in a way which is not conceptual, or ‘objects’ have to be defined in a way in which their ‘existence’ allows for ‘conceptual objects.’ And if we are defining ‘existence’ to include ‘conceptual objects,’ then we’re not talking about existence in the normal sense; What we are really doing is using our original definition of objects, as things what we can think about. We can’t drag all abstract objects, including Peter Pan and Santa Claus, into the larger set of objects that ‘exist’ if we want ‘existence’ to refer only to things which are not entirely conceptual.

    And if the most important and functional definition of an object is that it’s “something that we can think about,” Then we really have to question the motives for inventing objects which are both ‘abstract’ and ‘mind independent.’ What seems most likely is that we invented the term ‘abstract’ to refer to things that we can think about but which are not concrete, ie, physical, because a lot of the things that was think about only exist in our heads, like Peter Pan. It would be miraculous if this category that we invented to talk about peter pan also had deep significance about the fundamental nature of reality. Though it might.

    So let’s return to the axioms.

    1) Mathematical objects exist

    From the reasoning above, we can be almost sure that what this means is, “We can think about math.”

    2) They are abstract

    From the reasoning above, we conclude that this either means, “we can think about math, which is in our head,” which goes without saying, or “we can think about math, but it exists in a way which is not in the physical world that we know and also not in our heads.”

    What’s necessary here is that somewhere we must divide the types of objects into being objects which exist in our head, and objects which exist outside of our head. I think that the most straightforward way to do this is to immediately define objects as being things that we can think about, and then to split them into things which we can think about and that don’t exist in reality, and things that we can think about which do exist in reality. I think that mathematical Platonism makes the mistake of starting off by defining objects as things that exist in reality, and then slipping in the division of objects which exist in or out of our heads after we’ve already made an arbitrary and otherwise meaningless distinction between objects which are ‘abstract’ or ‘concrete.’

    Now let’s consider this from a slightly more mathematical perspective. In ZFC. According to ZFC, numbers are best described in terms of sets, or abstract groups of mathematical elements which have definable relations between them. One of the founding assumptions of ZFC is the concept of the set itself. A “set” is just a defined group of elements, such as the natural numbers. In mathematics, a ‘set’ corresponds to what we’ve been been calling an ‘object’, except that ZFC makes no efforts to differentiate between which ‘objects’ are real and which objects are not. It’s simply concerned with providing standards with which to define the relationships between sets, and once those standards are provided, mathematicians are free to use ZFC to describe whatever they would like within those standards. But if we’re trying to figure out whether mathematics is referring to something real, such as abstract, mind-independent numbers, and it’s correct to associate the idea of the ‘set’ with our statements about objects, then it must be that ‘set’s are referring to objects ‘which we can think about,’ and not necessarily objects which, ‘exist’ in the sense of being non-conceptual. The logical output of the alternative being that Peter Pan exists. It’s essential to the foundations of ZMC to assume that sets exist; this is because ZMC is a description of mathematics, which is an intellectual endeavor, and therefore, all mathematics is referring to ‘things that we can think about.’ It is not necessarily referring to ‘things that exist.’ Just like our definition of objects, some objects do refer to things that exist, but that is only a subset of all objects. All sets refer to things that we can think about. Only a subset of all sets describe things that exist.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s