Are there any features of reality that are necessary? That’s such a big, vague question you’d be forgiven for thinking it’s meaningless. The point of this post is to explore one meaning this question might have, associated with the philosophy of Immanuel Kant. Having done so, I will use Kant’s theory to test the philosophical coherence of a Victorian-era work of proto- science/speculative fiction, Edwin Abbott Abbott’s Flatland.

Scientists and others taken with Flatland have tried to develop its premise in a realistic manner, noting problems with Abbott’s portrayal, and suggesting what two-dimensional physics, biology, and engineering, among other things, might look like (A.K. Dewdney’s 1984 book Planiverse, for example, gives us a picture of what 2-d science might look like).

As far as I know, though, no one has tried to develop a philosophy of a world of two dimensions. I begin this project here by considering a famous argument that people claim to find in Kant and using it to show, perhaps surprisingly, that the world of Flatland is incoherent: it portrays self-conscious creatures moving around an external world but, I’ll claim, any creature such as described by Abbott wouldn’t have self-conciousness: experience would be, for them, an incoherent buzzing of sensation.

First I will present the basics of Kant’s philosophy, before considering the famous argument mentioned above. Then I’ll say some more about the universe of Flatland before showing how that argument applies to that universe.

This post is long: if you’re just interested in Kant you can just read the Kant and/or Strawson sections. If you know Kant and/or Strawson you can skip both, and if you’re familiar with Flatland you can skip my explanation of it.

Kant: How Are Synthetic A Priori Judgements Possible?

[T]he objective validity of the categories, as a priori concepts, rests on the fact that through them alone is experience possible (as far as the form of thinking is concerned). For they then are related necessarily and a priori to objects of experience, since only by means of them can any object of experience be thought at all.
The transcendental deduction of all a priori concepts therefore has a principle toward which the entire investigation must be directed, namely this: that they must be recognized as a priori conditions of the possibility of experiences (whether of the intuition that is encountered in them, or of the thinking). Concepts that supply the objective ground of the possibility of experience are necessary just for that reason.

So many words! You could be forgiven that it’s too technical and difficult to be worth caring about, or to be possibly interesting.

But, in fact, dejargonified his philosophy is one of the most interesting, staking out — something very seldom done — a genuinely new position and offering many intriguing arguments for them.

In order to appreciate what Kant was up to, consider this: reality appears to contain within it two very different types of truth. On the one hand, there are contingent matters of fact, like the fact that there is a packet of dried chickpeas on my bed right now. These are particular and could have failed to be: we can easily imagine that there was no such packet on my bed; it’s easily possible. Moreover, such facts can’t be known in advance and require some investigation. You need, in particular, to take a look at my bed to see how things are with it to assess whether it’s indeed the case. Call a truth like that, something contingent and depending on investigation to be learned, a posteriori.

Some truths are not like that, though. To use the tired example, that the bachelor who lives two doors up from me is unmarried is, indeed that all bachelors are, unmarried doesn’t seem like it could easily have been false. If we try to negate it, we get ‘some bachelors are married’, and that is necessarily false. Moreover, that all bachelors are unmarried is something you can know without inspecting a bit of reality. Call such things, necessary and knowable without looking at the world, a priori.

Now, there’s a possible explanation for this distinction. You might notice that the very definition of ‘bachelor’ includes in some sense being an unmarried man. And you might note that the definition of ‘a packet of chickpeas’ doesn’t in some sense include ‘being on my bed’. If you look in a dictionary under ‘bachelor’, you’ll read something like ‘unmarried man’, but no matter how much you root around the dictionary entries for ‘packet’ and ‘chickpeas’, you won’t find anything to do with being on my bed. Call truths like ‘all bachelors are unmarried’, which are in some sense definitional, analytic; call the rest synthetic.

OK, so we have two distinctions, each with two parts. You might be tempted by the following view: that all and only analytic truths are a priori, and all and only synthetic truths are a posteriori. Think about for that a moment; it should seem at least prima facie plausible.

Kant’s big thought, though, is that this isn’t so: in particular, not all a priori truths are analytic. There are synthetic a priori truths, truths that can be known in advance and which hold always, but which aren’t definitional. Examples are furnished by maths and geometry: that 7+5=11 is synthetic a priori, as is that a straight line is the shortest distance between two points.

What is particularly interesting about these truths is that they seem out there, in the world. Kant thought that Euclidean geometry was true of the space we inhabit, and so he thought that that claim above was true of the space we inhabit (we now know that the geometry of our space(time) is something only discoverable by investigation, and that there are geometries on which the shortest distance between two points isn’t a straight line. Let’s completely ignore these subsequent developments so as better to appreciate Kant on his own terms.) It’s puzzling how we could know what reality is like, in advance of looking at that bit of reality.

Now, even jargon aside, Kant’s puzzlement should reverberate. Reality does seem to have this weird bifurcation. If you drop a cup, it will always drop with the same acceleration(ignoring air resistance). Always. The dropping and the falling-with-that-acceleration are always connected. Some (indeed most) bits of reality, though, don’t have this very close always-y connection. The bit of reality of food being presented to my cat is not always followed by the bit of reality of him eating it. Usually, certainly, but sometimes he is sick (he has triaditis because his previous owner fed him chocolate — do not feed your cat chocolate!) and when that’s so, he’ll ignore food.

We should be puzzled by this, by the fact that the world sometimes has this alwaysyness to it. What’s the deal with that?

That was one of the main questions Kant wanted to answer. In order to get his answer, we need, alas, at least one more pair of opposites:

Intuitions vs Concepts

Now recall our big question: how can there be synthetic a priori truths? And let’s concentrate on the geometry case. The shortest distance between two points is a straight line. Here is something we (or at least Kant, pre-non-Euclidean geometry) seem to know. Any intuition we have, once we’ve applied concepts to it, will yield a picture of the world on which this truth holds.

What explains this fact? After all, there are many truths that are not built in to all intuitions. One possibility is this: maybe the very structure of intuition itself guarantees these truths of geometry hold.

Consider this analogy: we are watching something like a game of pinball. There’s the ball shooter-upper, and five holes at the bottom in which the ball can fall, but it must traverse the various shoots and traps of the machine. You notice that every ball falls in the same hole. If this happens enough, you’ll come to think, eventually, that the structure of the shoots and traps is such that the hole the ball always winds up in is the only hole it can possibly wind up in: the others are blocked off in some complicated way.

We could say something similar here: the alwaysness of this geometrical truth is guaranteed the structure of intuition. Intuition is not just a buzz of sensation, but has an order and articulation.

Okay — but remember the other bit about the a priori. It’s not only that these things always happen, but that we can seem to know this without inspecting reality. If you learn the notion of a point, and the notion of a line, then you can see in your head that it’s always true. You don’t have to go and measure actual lines. In our Kantian idiom: how can you know in advance that intuition has that structure?

And Kant’s big idea is that you can know this if you’re responsible for giving it its structure in the first place. The structure of intuition is something that we bring to intuition. Better put: intuition is not just a question of the external world impinging upon us, but is rather the product of the external world and our mind.

What that means is that what experience gives us, even before we’ve applied any concepts to us, is already partly something we have produced. Even before I apply ‘packet’ to the intuition I’m presented with, I have shaped that intuition.

Philosophy prior to Kant had swung back and forth between realism and idealism. According to realism, the world is out there independently of the work of our mind. One could also imagine an extreme form of idealism according to which reality was entirely the product of our mind. God’s mind, for example, might be like that.

Kant has a new position in between those two. Reality is partly independent of the mind (we are not responsible for the redness and whiteness and so on) but partly the product of our mind (we are responsible for the general spatial form that all intuitions must adhere to). This is his transcendental idealism, and is, as I say, a genuinely new position in the history of thought. And while it might sound somewhat out there, and hard to sympathize with in light of developments of non-Euclidean geometry, I hope you can appreciate that the problem it is responding to — the fact that reality sometimes has this weird alwaysness — is a good problem to worry about, and that the solution is interesting.

Fast-Forward A Couple of Centuries

It is here that I leave the historical Kant. He does have an answer. He presents it, in part, in what is called his ‘transcendental deduction of the categories’. This is the hardest part of the Critique Of Pure Reason, which is one of the hardest books in philosophy. Books have been written about it; careers formed based on just studying it.

I’m not going to try to explain it, because I don’t understand it myself. Rather, I am going to consider how a roughly similar question was asked and answered in the late twentieth century by Peter Strawson. Once I have done that, we will have the tools necessary for examining the philosophical cogency of Flatland (it should be noted, that all that stuff I just said about intuition and idealism won’t figure in my discussion of Flatland. I just mentioned it because it’s independently interesting).

To see Strawson’s view, note that our picture of the world is as so: there are self-conscious agents who exist in a world that exists independently of those agents. To save time, let’s say that our picture of the world is that of a coherent world, where ‘coherent’ means: a world consisting of self-conscious agent who exist in a world that is independent of those agents.

But what does this mean? What is self-consciousness and what is mind-independence? If we can answer these questions, we might be able to understand the fundamental necessary concepts that make up our picture of reality.

Strawson’s strategy (in the book Individuals, and in particular in its chapter 2) is to approach this question by considering incoherent worlds and coherent worlds other than ours, to see what features they lack or have, and on that basis to try to specify some generalizations about what any coherent conception of reality must be like.

It turns out, it’s not so difficult, and very interesting, to do so. Thus consider the world presented in this thought experiment:

Chaotic, Motionless. This is a world in which the only sense is sound. You hear sounds, but according to no order or pattern: now a G, now a B, now a C, another C, a G again, an F, an A, and so on. A random collection of notes goes on forever.

Here is a plausible thing to say: such a world seems incoherent. If you were dropped in it, your experience would be a mere buzzing confusion of which you could make no sense. To help convince you of this, consider:

Ordered, Moving. This is also a world with only sound. There is, however, unlike the last world, a master sound that is always heard, but whose pitch varies. One’s experience consists of ‘moving up and down’ the pitch dimension,which is to say that for any two consecutive experiences e1 and e2, either the master pitch of e1 is the same as the master pitch of e2 (in which case it’s as if one is stationary in the pitch dimension), or the master pitch of e1 is very slightly higher or lower than that of e2 (in which case one is ascending or descending the pitch dimension). But now imagine one starts at lowest pitch; one hears a G. The pitch remains, one hears a G still; one moves up in pitch, and hears a B, up again and hears a C. Then one moves down two pitches and hears a G again, and in general, whenever you’re at a given master sound pitch a given note always plays.

This very simple world, arguably somewhat surprisingly, is much more coherent than the first one we considered. To see this, note that we can make sense of mind independence, in particular we can make sense of sounds as existing unperceived. The agent of such a world can think, when they are at pitch 3 (thus hearing C), that right now, at pitch 1, G exists and can be heard. That is, G is independent of the mind because there’s reason to think that it exists when not perceived. Think back to Chaotic, Motionless: it doesn’t seem that that’s a thought one can even begin to formulate in the world. The question of whether two sounds are one and the same (as opposed to being of the same type) is not a question one can sensibly ask in such a world.

In a similar sort of way, a being in Ordered, Moving might be able to have self-consciousness, because they can make a distinction between states of the world, and states of themselves. One way to note this is that they would have room for the concept of hallucination, where a hallucination is a state of the subject that doesn’t correspond to a state in the world. In particular, imagine our hearer to inhabit the world for a long time, and, after many trips to 1, on one occasion they return and hear not G but F. They then move to 2, hearing B, and return to 1 and hear G. Then it seems plausible that the agent would think that their previous experience at 1, when they heard F and not G, was a hallucination, because it’s a feature of the structure of their world, as they understand it, that G is at 1.

In general, this world opens up a distance being what is presented and to whom it is presented, and this distance, arguably, would suffice for the agent to realize that it is a part of the world separate from the sounds it experiences, and this might suffice for self-consciousness. Again, it seems that we can’t make the same sort of distinctions for Chaotic, Motionless: we can make little sense of hallucinations in such a world.

Our two worlds seem to differ in coherence. So let’s ask: what is special about Ordered, Moving? Well, it has two features. Firstly, it has an underlying order or pattern (exercise for the reader: consider Ordered, Motionless. What are we to say about such a world?) But secondly, it has the notion of space: the pitch functions, in essence, as a space in which objects can reside, and that allows for them to exist unperceived.

This idea, I take it, is central to Strawson’s views about coherence: for a set of experiences to be coherent, it is necessary for them to present something like a space, because space enables one to think of existence unperceived. My eventual strategy will be to argue that Flatlanders won’t be able to latch onto a concept of space, and so their experience will be incoherent.

The front cover of the book, from Wikipedia. Citation info is apparently as so: *EC85 Ab264 884f, Houghton Library, Harvard University.


Flatland is a two-dimensional world, and the objects in that world are two-dimensional figures. Imagine an infinite piece of paper, on which are drawn squares, equilateral triangles, isosceles triangles, straight lines, and other such figures. Some of these figures are people, at least two-dimensional people, while some are inanimate objects. Thus if we imagine zoom in on a part of our page containing a pentagon and a square, the former might be a house — all houses are pentagons, although not vice versa — while the former might be a ‘Professional Man or Gentleman’.

Persons have some of the same features as we do. They have the power of locomotion, for example, and the ability for sense-perception and memory (at least, they’re meant to — as I said, I’ll be arguing the story is actually incoherent), in ways that I will get to. They differ in intelligence and social class and education.

This latter feature is one of the enjoyable parts of the story, so, at the risk of spoiling it for other readers, let me note that in Flatland, one’s social class and education is entirely determined by the shape one is. In particular, the more sides one has, the higher class/intelligence one is. (It’s part of Abbott’s satire that no distinction is drawn between class and intelligence, a fact that is meant to be reflective of how Victorians viewed the world). Squares, for example, are higher class/intelligence than equilateral triangles, who belong to the middle class. The working class are isosceles triangles with two long equal sides and a very short bases (so that they are almost lines). Moving on up, priests have so many sides that they are essentially indistinguishable from circles. Women are straight lines, a reflection of the low esteem mainstream Victorian society held them in.

A consequence of all this, of course, is that social mobility doesn’t really exist. Since one can’t (typically; it’s allowed at certain places but it’s not a central part of the picture) change one’s shape, one is stuck with the intelligence and social lot in life of one’s class.

(One might ask: are we, in this respect, Flatlanders? We might think obviously not, that we allow people to rise above their station via education and diligence. But, well, one’s fate in life is to a large extent determined by the circumstances of one’s birth. You might even think that were our world like the Flatlanders’, we would be more aware of such facts and might strive to fight against them. As it is, because someone brought up in a deprived area and someone brought up in a posh area might look more or less the same, we might be inclined to think that the differences in success are functions of who they in particular are, and not of features of their history. Flatlanders’ history is etched in their figure, and such mistakes wouldn’t be so easily made there.)

Anyway, this is the first important point: in Flatland, many people look the same. There are many working class people, many middle class people, many priests, all moving about the place.

Let me now say something about their capacity for perception. It is interestingly also divided along class lines. To begin to see this, note that while Flatlanders have eyes, visual perception isn’t that useful. Imagine one is a Flatlander. The easiest way to do this is to put, say, a coin, on a desk, and gradually position oneself so that instead of looking over at its face, one is looking across the desk at its side. At a certain point, the coin will look like a straight line.

The same applies for all figures: if one looks at it side on, it will look like a straight line. This means that vision is more or less useless as a way to discriminate objects. Seeing a figure before one, thus a line, one won’t know if it’s a circle — thus deserving of respect — or a workman, or even — heaven forbid — a woman.

But respect is important in this world! You need to know the status of the people you bump into. Thankfully, if vision doesn’t allow this, touch does. All Flatlanders are regular figures: their sides are the same length. This means (well, Abbott seems to think it does, but Ian Stewart, whose annotations are useful, informs me he’s mistaken) that their angles are the same length. And this means, in turn, that you can feel one angle and thus determine the figure of the creature you’re beside: if it’s 60', then it’s a middle class equilateral triangle, if it’s 90', it’s a professional square, and so on.

So that’s what they do. Touch is more important than vision for determining the class of the person that you’re in contact with. This is true for everyone but the upper strata, who in fact are able to rely on sight. I won’t get quite into the details but Abbott sets it up so that with long mathematical training, one is able to figure out, on merely looking at a line presented to you, what figure it represents, roughly by seeing how fine or distant different parts of the line appear to you. The point is, though, that also for those with long mathematical training, perception only gets you the class of the person whom you encounter. It does not let you distinguish between members of the class. That’s very important: keep that in mind.

Is It Coherent?

(That said, it’s yet another interesting thought experiment to imagine what such a society would be like, in which one could only differentiate between people as to class. Would it be one in which we had a right to treat all people of the same class as the same? If I needed something built, could I pay any isosceles workman to build it? If I had a lawyer son, could I treat any equilateral triangle as if it were he, inviting them to lunch and such? Would no such personal relations exist? This is, I think, a very interesting question, whether such a world would be, so to speak, socially coherent. But it’s a question for another time.)

That’s not so, though. Although Abbott doesn’t make this move, one could, from such a setup, introduce particular recognition. You could do it as so: each person assigns themselves a name, which they write in some angular script on a piece of wood and then carry around with them. In order to recognise someone, you feel the piece of wood, thus reading the name tag and working out the identity of the person.

(Another digression: you could extend this idea by letting people write not only their names, but also their life story on the piece of wood, and let people communicate in the same manner. In such a case, Flatland would ironically quickly change from an exceedingly superficial society concerned only with status to an exceedingly unsuperficial one. If one were deciding whom to marry or befriend, for example, one wouldn’t do it based on looks, but based purely on information the person reveals about themselves in writing. In a sense, augmented with some writing device, Flatland would have some of the same features as digital worlds where people bond based (sometimes) only on information, and not on appearance or other physical attributes of people.)

Returning to the main line of argument: although one could introduce particular recognition via writing as suggested above, that would mark an evolutionary step forward for the Flatlanders, who would, before being able to make that step, have had to already been able to deal with their world as described in the book, devoid of particular recognition. My claim will be that they won’t be able to get to the first step of having such a world to make the necessary further step.

The Argument

Now, when we move to actual spatial space and away from the Strawsonian sound space, we need something to replace the changing pitch that can serve to mark location. But that’s easily found: in actual spatial space we rely, typically, on landmarks to distinguish one place for another.

One can see this by noting the great difference between an imaginary spatial world that consists of endless homogeneous green fields and one in which landmarks regularly break up the fields: a stonehenge here, a hill there, a lake there, and so on. Put in the latter sort of world, one could develop a conception of space and mind-independent particulars; placed in the former, it’s arguable that one could not.

Now Abbott doesn’t say much about the natural world in Flatland, so let’s assume that it also consists of figures pretty much like the people (what else could they be?) For simplicity, assume the only natural material is wood, and its angularity reflects, say, quality of grain. Then we could imagine a natural world like the following, where all the figures are different types of wood:

A part of Flatland space, with landmarks of different shape.

This would be a space somewhat akin to Ordered, Moving. A creature could be at the square wood (bottom left), and move around the edges of the space clockwise. After a couple of turns, they would begin to recognize a pattern: they could say, for example, after passing the hexagon for a third time, ‘I know, the square is coming up!’ And when they were at the square, they could then think ‘If I were to move back, I would see (feel) a hexagon’. Similarly, they could make sense of hallucinations, and thus of the possibility of what they sense coming apart from what’s out there in the world.

All good then, it seems: the Flatlanders, were their natural world so arranged, would be able to get a concept of space. The problem — as so often in life — is people. Because remember, Flatland contains people moving about, meeting each other, and so on. So we need to add figures representing the people. We might have the following (where colour is used to help us distinguish people from wood. Note that Flatlanders would notice no such difference):

That same space with some Flatlanders in it

Now imagine starting at, say, top right. You feel the triangle then the square. Remember, there’s no difference in feeling between inanimate and animate objects, so for all you know, that’s either two pieces of wood, two people, or one of each. What that means is you really can’t form a hypothesis of what it is you’re feeling. You move clockwise, and assume that other creatures move counterclockwise. You then feel the hexagon and the newly moved equilateral. Again, you can form no hypothesis as to what exactly you’re feeling. You’ll be pretty confused.

But the big problem comes when you go back on yourself. Before, at top right, you had right-triangle and square. But there’s a very good chance that you won’t feel a right-triangle and square again. You will only feel a right triangle and square if the middle-class square moved first counterclockwise and then clockwise. But there’s no guarantee that this will be so: it could have moved in any number of ways (or, indeed, stayed still).

This will generally be so: as long as we assume that creatures move to and fro in no set order, as creatures with minds are wont to do, then in light of the fact that creatures and inanimate objects conceived of as location-markers are the same, you will never be able to associate a fixed constellation of shapes with a particular location, and so you will never be able to get the concept of objective space. It will just seem patternless, and if it seems patternless, then your experience will be chaotic, because we took from Strawon the claim that a conception of space is needed to underwrite a coherent objective world. So I conclude, under these assumptions, that Flandlanders won’t be able to form a conception of an external world, and I take it that this shows that the Flatland world, containing as it does self-conscious creatures who take themselves to belong to such a world, is incoherent.

Which, I think, is pretty cool. Previous work and commentators have pointed out ways in which Abbott’s two-dimensional world is physically, for example, problematic. This shows that it is also philosophically so.

Two Codas

But I want to end with two points. The first is something that I have been withholding from you, reader: sound. Sound features in Abbott’s world, and some of the things he says suggests that it might offer a way out of the puzzle I’ve proposed. At the same time, it’s not clear that what he says is incoherent, and that’s why I didn’t discuss it.

So, Flatlanders can speak and can listen. The lower classes can recognize my sound. From chapter 5:

[One] means of recognition is the sense of hearing; which with us is far more highly developed than with you [i.e. we Earth dwellers], and which enables us not only to distinguish by the voice our personal friends, but even to discriminate between different classes

The lower classes can hear shape! But that’s not the interesting thing. The interesting thing is the suggestion that they can distinguish personal friends. This implies, surely, that the voice can discriminate between two objects of the same shape: if your friend is equilateral, and there are other equilateral triangles, and you can recognize him by voice, you must be able to distinguish between equilaterals by voice.

The problem is that this only works partially. It only enables us, seemingly, to discriminate between beings of the lower class. A little further on we read:

As we ascend in the social scale, the process of discriminating and being discriminated by hearing increases in difficulty, partly because voices are assimilated, partly because voice-discrimination is a plebeian virtue not much developed among the Aristocracy

You might think voice could help solve the puzzle as so: one of the sources of the problem was the inability, in essence, to distinguish between landmarks and people. You’d never be able to get a sense of a landmark because you’d never be able to separate it out from the constantly differing people around the landmark. But if the voice is available, maybe you could make that distinction.

Maybe. I’m not sure. Could the voice work like this, especially in light of the fact, per the second quote, that voice recognition is limited? I’m not sure, and leave it as an exercise to the reader (an intriguing possibility is that because the lower classes can voice-recognise, it could be that they but not the aristocracy could have a conception of an external world, which would be fun.)

Finally, let me end with a last, and somewhat tongue in cheek, argument. Prejudice, at least in part, arguably, is seeing all of a class of people as the same: the racist views all people of the race they dislike as the same, the misogynist thinks, of women, “they’re all the same”.

The Flatlanders are extremely prejudiced: social class is quite literally all they see. And the Flatlanders, if you buy my argument, won’t have a coherent conception of reality because of it.

Maybe we should say the same of the prejudiced in our world. That, to the extent that they do view different people as essentially the same, they are fundamentally confused as to the nature of reality and unable to navigate it inappropriately, seeing similarity where distinctions are. Their world, you might think, has a chaoticness at least similar in kind if not in degree to, say, our Chaotic, Motionless sound world presented above, or to the Flatlanders’ world. And so we have an extremely roundabout, weird argument against being prejudiced: it makes reality incoherent.

That we can and should ask about the scientific accuracy of sci-fi and speculative fiction works is commonly accepted. If you buy the project of this post, we can say the same thing for philosophy, and we can learn about fiction by bringing to bear philosophical arguments and theories, which is, I think, both interesting and cool.

Novella "Coming From Nothing" at @zer0books ( Academic philosophy at: