The Mathematics, Metaphysics & Logic of Absence

I want to know if a number exists when there is no one around to count it. Furthermore does thinking about a number, conceiving of it, imply that it exists? Or is a number that is not associated to a set of actual objects just a chimera, where we concatenate distinct and separate impressions. Like piecing together a mystical animal out of the body parts of animals that actually exist. So if we say that we believe that a number exists when not associated with an actual object, our statement is equivalent to stating that we believe a unicorn is real simply because you can conceive of it?
If that is the case, well then I suppose that would imply that numbers are more than just linguistic, symbolic structures. It would also mean that we live in a world inhabited by unicorns and fairies.
I’m no mathematician, but I look around me and something tells me that the implications of our previous assertions don’t appear to have much bearing on reality.

To be honest, I have never understood mathematics. I can use a trivial amount of it, but I hardly understand it. In truth I think that some of the fundamental axioms that underpin mathematics smack of the meta-physical and the mystical. But I assume there is just an awful lot I don’t understand. So humour me while I try and figure out the basics on my own here:

If we enumerate 1 and then 1 then we get 2, correct? But what is that 2 supposed to be representing, two real world objects or two numbers? Are the symbols referring to two chickens or are the numbers referring to themselves? The later proposition doesn’t mean anything as far as I can tell, because 1 apple explains something. We have placed on that apple the attribute of being 1 of something generally described as an apple. But 1 number 1, explains nothing; how could a number explain anything when it is just referring to itself.
So unless a number is an extension of an object then it is basically a self-referential statement, which has no value beyond it being a linguistic construction. Meaning the purely abstract number 1 is essentially an empty set ({}) with the potential to hold and be attributed to 1 object, 111 to one hundred and eleven objects and so on.

I feel like I am probably mixing up different fields of mathematics, but I don’t see a problem as of yet. Sets are groups or numbers which represent objects, so that {Apple1,Apple2,Apple3} represents the number 3 or the statement, ‘I count three apples’. That seems to make sense, but what if we start filling sets with purely abstract numbers. Numbers we have already decided represent empty sets when not attributed to and thus holding instances of objects, so that: The number 3 can be represented by { {},{},{} }, which is a set of three potential objects. This number 3 could then be simplified by just indicating the absence of objects in a set by saying {}. But we already said that the number 1 or any other purely abstract number can be represented by {}.

What does that all mean? Well I don’t know honestly, as I said I am just trying to figure it out as I go along. I am sure there are children out there with a better grasp on this stuff than me, but let’s just keep going and see what conclusions are drawn.

I suppose that the most obvious implication of this logic would be to say that, any number that is not an attribute of an object or set of objects, is essentially representative of the absence of value and meaning. Represented by {}, which is essentially a placeholder for absence as true absence is beyond our comprehension.

That would mean that unless there are an infinite number of objects in the universe, then there cannot be a meaningful use for numeric infinities. As once the last object has been accounted for in the universal set, once you add an empty set of 1 to that, you lose all meaning and you end up with a rather large number representative of nothing, which can again be represented as {}, or if we want to be more accurate we just wouldn’t refer to it with a symbol at all.
Essentially, you can’t just add one to the highest number you can think of, because unless that number is representative of distinct and unique objects or collections of objects, then it is meaningless and thus represents absence.

Another result of this logic would be that all distinction of objects as unique and definable as atomic instances of a certain type, namely an ‘apple’, is completely arbitrary. As ‘Apple1’ and ‘Apple2’ could be different sizes, density and more than likely have more or less atoms in one or another of the two apples. Yet we put them in the same set of objects, {Apple1, Apple2}, and represent that with the number 2. Not a particularly accurate measurement and furthermore, we have just shown that counting is merely a mental construct, wherein we generalise and group objects together. My point being that if there is no entity around to do the counting, then numbers don’t exist independent of thought. It seems that numerical infinities are contingent on the existence of an entity capable of counting for an infinite length of time. So even if an infinite number of objects do exist then numerical infinities are still in doubt.

I agree that something doesn’t sound right here. But common sense is a biological constraint and it would be foolish to image that the universe should be consistent with what seems ‘right’ to the human race.


A Critical Response to The Art of Game Design: A Book of Lenses

Creating Games

It isn’t often that I get to read and respond to a piece of writing that I disagree with on so many levels. So to say that I didn’t enjoy detailing my critical response to Jesse Schell’s, The Art of Game Design: A Book of Lenses (Schell, 2008) would be a lie.

The first of the two chapters that I will replying to is Ch. 10: Some elements are game mechanics (Schell, 2008), in which Schell divides mechanics into the following six elements:

  • Space
  • Objects, attributes, states
  • Actions
  • Rules
  • Skill
  • Chance

The second of the two chapters being Ch. 15: One kind of experience is the story (Schell, 2008). I won’t address this chapter directly, but I will refer to its main tenants throughout my response to the former chapter.


Mechanic 1: Space

The mechanic of space according to Schell is a mathematical construct. This is a reasonable definition that just so happens to be one word too long. Space within a video game can be thought of as a construct, but in the case of a narrative driven text adventure, space is purely abstract.

I assume Schell’s argument stems from computer science originating from mathematics. This being the poorly expressed rational as to why every video game is a mathematical construct at root according to Schell.
But if we follow this same thread of logic, it also appears to imply that as books are now composed on and printed by computers. A computers mathematical basis implies that the spaces expressed in a physical or digital copy of Franz Kafka’s, The Trial (Kafka, 2011) are also mathematical constructs – which is obviously false!

You may be shaking your head at this point, saying to yourself ‘No Tom, that doesn’t make sense’. But I assure you that, Schell’s supposition making no sense is of no fault of mine. All I have done is take the idea seriously for the sake of testing it for logical consistency, a test which it has failed.
But if we look objectively at Schell’s supposition we can see the error. That being that there might exist an a priori conceptual ‘space’ within the mind of the designer, before it is subsequently expressed through a particular medium and in a particular language. Be that through the use of computer rendered images, spoken word, written word, mathematics, abstract logic or your own special notation composed of twigs and bent spoons – whatever!

This may seem like a trivial distinction, but the statement that ‘…as a game mechanic, space is a mathematical construct’ (Schell, 2008) is both incorrect and ultimately limiting.


Mechanic 2: Objects, attributes and states

This section begins with Schell formulating the following self referential sentence, which as far as I can tell, means absolutely nothing:

‘A space with nothing in it, is well space’ (Schell, 2008).

In explaining what space is, you can’t refer to the concept in question as an answer. It would be the same as me saying that:

‘A forest with nothing in it, is well a forest’.

In truth, a forest without anyone in it to observe what the forest is, may or may not exist in any number of configurations, some of which may not be a ‘forest’. It is a completely arbitrary choice on your part if you choose to believe in what you cannot possibly know.
I feel the least that Schell could have done here would be to not make unqualified statements as if they mean something, when in fact they do not and cannot mean anything.

What is the lesson here? Well firstly, self referential sentences can’t be used to define anything useful within the langauge the ‘closed loops’ have been formed in. The second lesson being that you can’t just say things because they sound credible, they also have to make sense. Especially when you are setting out the axioms for your model, otherwise everything else is undermined – It is like building a house on imaginary concrete!

At any rate, this acts as a poor segue into the discussion of objects, attribute and states. These all being topics that are covered in brief, but without any great error. That is until such time as we reach the topic of ‘Secrets’, a concept also known as ‘perfect’ and ‘imperfect information’ belonging to Game Theory. Schell having previously dismissed Game Theory as far too limited in scope and application to be relevant to modern video games. It must be said that Schell rebranding this well know concept looks rather silly. It also calls into question the extent of Schell’s knowledge on the subject. Especially given that Game Theory was so quickly dismissed, yet we find one of its main mechanics has been subsequently appropriated without reference within Schell’s own system.


Mechanic 3: Actions

The most interesting thing about this section is that Schell has moved from describing space, objects and states in terms of mathematical constructs and now begins refering to actions and their related states in terms of a linguistic metaphor. In this metaphor Schell describes various types of actions as ‘verbs’.
This is a very interesting and powerful concept for understanding video games. It is also expressive of a certain level of cognitive dissidence that must exist within the author. But aside from everything else, I must agree with Schell statement as follows:

‘[Creating verbs that can act on many objects]. This is the single most powerful thing you can do to make a game elegant and interesting’ (Schell, 2008).

But my agreement ends at the point Schell attempts to tie the dialectical knot by adding the following:

‘Games usually limit players to a very narrow range of potential actions, while in stories the number of possible actions that a character can engage in seems nearly limitless. This is a natural side effect of the fact that in games, that actions and all of their effects must be simulated on the fly, while in stories it is all worked out ahead of time’ (Schell, 2008).

This statement is incorrect in that it denies that when we read a book, our mind must simulate for us the environment, objects and narrative that we are reading about. So Schell’s statement is not only contradicting itself from one point to the next, but the message Schell is attempting to drive home isn’t even remotely true.

Perhaps what Schell was trying to express was the difficulty that exists in adding elements of interactivity to a game without detracting from the narrative and disrupting the simulation. A very simple way to explore this so called ‘action gap’ (Schell, 2008) is to look at one of the original Fighting Fantasy novels.

Here we have a non-linear narrative containing both ‘discrete’ and ‘abstract’ space. As well as objects, states and more verbs than you could poke a stick at. Combine all of these elements and the result is an unimpeded narrative, which also happens to be an interactive game. I say ‘unimpeded’ in that it involves only linguistics, small amounts of arithmetic and a human brain to perform the simulation – basically in this case we arn’t limited by graphic or sound quality in experiencing the simulation. Let us take Deathtrap Dungeon (Livingstone, 2009) as the perfect example of this kind fo game – I am assuming that anyone with a minor interest in games will know the format, so I won’t go into detail.

Deathtrap Dun-01-01


Now if we consider a classic text adventure, such as Leather Goddesses of Phobos (Infocom, 1986). Here we have a non-linear, narrative containing both ‘discrete’ and ‘abstract’ space. As well as a number of objects, states and verbs the extent of which is only limited by the available storage mediums of any given period of time.
The number of words that can be stored in the modern day is near enough to infinite as to not matter. But there is still an upper limit to the number of objects that any programmer could feasibly create. I believe this is the root of Schell’s argument, but was better expressed by Steven Meretzky in the documentary, Get Lamp:

“The total text in a typical Infocom game, maybe would equal like a thirty page novella… …you can’t handle everything that somebody might try. What you want to do, is you want to handle the most common things that people will try and the most interesting things that people will try… … [You have to ask yourself] how much of the very limited resources that you have do you want to expend on letting the player go down in that direction. So in a lot of cases what we just did in that case was just kill the player” (Get Lamp, 2010).

So the question I am left asking myself is, how is it that a game like Deathtrap Dungeon (Livingstone, 2009) succeeds where subsequent games that aspire to a similar level of emergence have all failed?

One reason may be that the details of all objects in a game of Fighting Fantasy are left partly to the reader’s imagination. You are given only a limited number of possible interactions with specific objects related to the narrative, which is to say the narrative is ‘on rails’. Whereas in a text adventure game you might walk into a room and say ‘Do a backflip’, only to have the game respond by saying that it doesn’t understand your command. Or in the case of a first person shooter, you might expect to be able to blow holes in the ground, but you can’t. In each of these cases the illusion of freedom is broken. But in the example of Fighting Fantasy, the illusion of freedom is kept intact as you are only given three options to pick from at any one time.
This kind if interactive fiction does away with the illusion of an open world, where you essentially have to guess what is, or is not possible. And it just says, ‘this is the narrative, you get some choices and there is a chance that you might make a wrong move and have to start again. Make of it what you will’ – Very simple.

Unfortunately for the text adventure genera in general, reading, literacy and the use of one’s imagination are seen as taxing endeavours by most people in the modern day.
So people continue to blunder around with poor simulacra and wonder why all of their shinny, lifelike computer rendered images continue to produce experiences which seems so limited.
Then it apparently follows that those people go off and make up terms like ‘action gap’, hoping no one will notice them attempting to square the circle.

I personally think that the secret to emergence may lay in crafting an interactive narrative and in creating a simulation that is unimpeded. Unfortunetly this isn’t a secret that I am a party to, so I have no answers, just speculation – lots of speculation.

This is also a nice point to quickly diverge for a short while from the current chapter, to the chapter entitled Ch. 15: One kind of experience is the story (Schell, 2008)Schell, interestingly enough seems to shift into writing what could only be described as a manifesto. Here Schell makes unqualified arguments touching on the apparently insurmountable challenges that, according to Schell, prove interactive storytelling to be a fool’s game and an apparent mortal ‘sin’ of game design.

Far be it from me to point out that a challenge in actualising a concept does not mean the concept has no value and may not be achievable in due time. As Schell seems to enjoy his mock-scientific analogies I may as well prove my point in a like manner:
Take the search for the Higgs particle. Yes, it was incredibly difficult to conceive of the concept. It was even more difficult to then build a machine capable of creating and detecting the particle. Again it was amazingly difficult to then process and analyse the results. But given plenty of time, money and energy the particle, which was previously beyond the scope of human experience was found.
There was in this case no assumption that just because the task happened to require processing power that did not exist at the time the idea was conceived, as a result, the idea must be misconceived. 

If we were to apply Schell’s own logic to this same issue, we would have to say the following: ‘As we can not know if the particle exists, the concept is not practical and therefor it is not true.’
It does not occur to Schell that just because something is too difficult now, that doesn’t mean it won’t eventually become both achievable and practical in the future. This makes Schell’s constant reference to those who believe in the credibility of interactive fiction as ‘Dreamers’ unnecessarily condescending.


Mechanic 4: Rules

Schell unwittingly provokes an interesting question when stating the following:

‘[rules] make possible all the mechanics we have seen so far and add the crucial thing that makes a game a game – goals’ (Schell, 2008).

Is this statement really true? What if I play the game and completely disregard the goals set out for me by the game designer. If I was to play Grand Theft Auto (BMG Interactive) and never once bother to follow the narrative or gaols, am I still playing the game?

Perhaps I am setting my own gaols and the simulated environment acts as a sandbox, within which I might choose to set these goals -That makes sense.
But now we are saying that, to follow any goal, be that to move from coordinate A to coordinate B within a simulated environment qualifies you as having played a game.
So be that simulated environment a chess board, a fully realized 3D world or a simple text adventure. In each of these cases you are both simulating an environment and setting goals, however trivial. This means that the only difference between a game and real life seems to be that we consider our everyday life not to be part of a simulation and so we feel comfortable drawing a distinction.

I would actually offer this up as a more satisfactory definition of what makes a game a game, rather than just saying ‘goals’ as Schell has.


Mechanic 5: Skill

The only call-out regarding this very short section is that Schell seems to have made an unnecessary distinction between ‘mental and ‘social’ skills. Perhaps this was done in order to make his point triune as is a habit in Western culture. Then again, perhaps Schell feels that there is a true distinction between the problem solving involved in manipulating and understanding people as opposed to anything else that might be ‘worked out’.
I don’t personally see this distinction as a valid one and Schell’s own focus and scope in this section occludes any need to make this distinction. So I can’t really understand why it was done, Schell is seemingly happy to let us speculate.
But how you choose to define your terms is always going to be arbitrary. I just don’t see this particular distinction adding any value to Schell’s model.


Mechanic 6: Chance

According to Schell, chance is ‘an essential part of a fun game’ (Schell, 2008), which is true. No game wherein you know the outcome every time is going to engage a player for very long. Unless of course it is a Poker Machine, wherein the ‘chance’ is controlled, meaning that it is removed. Yet people still play poker machines, presumably because they are having some kind of fun while doing it. So I guess Schell is incorrect on this topic also – I know very inteligent people who understand statistics and also understand that there chances are fixed, who love throwing money away in a casino.

I prefer the contention that chance is an essential part of a realistic game. This statement unlike Schell’s may actually be true. If a game is going to simulate reality (I should note that ‘to simulate reality’, may be a self referencing sentence, but I am going to leave it in here to prove a point), then it must simulate the randomness of nature and the uncertainty that is involved in any instance of cause and effect.

Apart from that one point, I would not disagree with any of what Schell says in this section regarding the usefulness of understanding probability. It is just interesting that my own point regarding controlling the probability in some games failed to occur to Schell.


In Conclusion

Schell seems to have a tendency to dismiss anything that appears to be too complex or impractical. If I was being unkind I might say this piece of writting reads as though it was written by a myopic pragmatis, but I’m not feeling unkind today. All I will say is that Schell’s perspective on video games has been clouded and ultimately limited by some confused logic.




Deathtrap Dungeon.jpg [ONLINE]. Available at: 27/04/2014].

Franz Kafka, 2011. The Trial. Tra Edition. CreateSpace Independent Publishing Platform.

Get Lamp, 2010. [DVD] Jason Scott Sadofsky, USA: Bovine Ignition Systems.

Grand Theft Auto, 1997 [Video Game] New Yorl City, New York USA: BMG Interactive.

Ian Livingstone, 2009. Deathtrap Dungeon (Fighting Fantasy). Edition. Icon Books Ltd.

Jesse Schell, 2008. The Art of Game Design: A book of lenses. 1 Edition. CRC Press.

Leather Goddesses of Phobos, 1986 [Video Game] Cambridge, Massachusetts, USA: Infocom.