[This is #4 in a series of posts exploring information: what is it, where does it come from, and what does it teach us about ourselves and the reality in which we live. – #1 can be found here]
When encoding information using symbols, it doesn’t fundamentally matter what physical carriers are used to display the symbols (ink on paper, light on a display), or what form the symbols take (shape, colour, size, pattern). What matters is that the system offers a vocabulary and a grammar with enough scope to define a set of concepts we need to communicate, and to relate them to each other in the necessary ways. If a notation system lets us specify objects / concepts distinctly using recognisable symbols, we are on our way towards being able to represent and transmit information.
For instance, we can consider a musical language system. This needs – most basically – a means of encoding ordered sequences of pitches, and defining a length of time for which to hold each pitch. This means can be provided by the familiar five-line stave notation and note alphabet of the Western musical system. But this system is not necessary for encoding music; after all, in previous ages, plenty of other symbol systems have been used. Ultimately, as we will see, we are free to use any encoding system with sufficient specificity to distinguish the tones and timings needed to reproduce melody.
Figure 1- a melodic phrase from Bach’s Sonata No. 1 in G-minor
For example, we would usually choose represent a phrase of music – for instance “G,B♭,G,D,G,D” – as a sequence of notes on a stave (Figure 1). In other words, we would use different symbol shapes (quaver, crotchet, minim etc) to differentiate the time value of each note or rest, and vertical position to differentiate pitch. Because we human beings can generally distinguish different shapes and vertical positions with our vision, we can use these visual differences to represent differences in time value and pitch. In other words, in Western notation, each of the different symbol shapes and their positions are specific enough to let us assign specific concepts to them (i.e. vertical spacings for differences in pitch and horizontal spacings for differences in time). This means instructions for producing melody – the actual information content of the visible score – can be represented graphically.
A new music notation system
But we could easily choose a different system and still enable the same specific encoding of different pitches and time values. For instance, in the system shown in Figure 2, different colours have been used to denote different pitches.
Figure 2 – a notation system for music built around colour and area instead of shape and vertical position
Provided we can tell the colours apart, and there are enough of them, we can assign a specific colour to represent a specific pitch. We can also represent time values in a new way. Instead of using different shapes of symbol (as in Western notation) we can use the same shape for all time values, and change the size instead. Thus, in the system shown in Figure 2, the note area is proportional to the time value of the note. So, for example, twice the area means twice the length of note.
This is all we really need for a basic music notation system: the ability to encode specific pitches and time values. The system in Figure 2 also adds a way of encoding octaves, enabling our limited set of colours to be extended to cover more pitches, just as ledger lines above and below the stave extend the range of pitches in Western notation. Figure 3 shows the phrase from Figure 1 (the opening of the 4th movement of Bach’s Sonata), transcribed into this new notation system.
Figure 3 – the phrase from Figure 1 transcribed into the new notation
It looks different, but contains the same information. Indeed, using the mapping shown in Figure 2, we could recreate Bach’s intended melody correctly from this new representation. (Another example is shown in Figure 4).
Figure 4 – another phrase of music transcribed into the new notation system
What fundamentally matters for communication of music, or indeed any kind of semantic information (such as the content of a play, a conversation or a poem) is the specificity of the signs in the vocabulary (i.e. that each sign is distinguishable from the others, so they can be mapped to different concepts), and that the signs can be freely arranged (e.g. in sequences, sentences or phrases) in order to hold freely-ordered information configured by a human mind.
The point of the musical example is simply to show that the same fundamental information content can be expressed using different symbol systems. Information is not ‘locked’ into a certain system of representation: it is not somehow a ‘property’ of a certain language. Rather, information itself pre-exists the linguistic representations we commonly use to communicate it. In other words, the encoding of information (i.e. the notation, the writing, the choice of grammar and vocabulary) is a secondary factor to the existence of information.
In music for instance, the popularity of the Western notation system over others is not because this notation is fundamentally ‘tied’ to the information content of the music it often represents. After all, a person can compose, memorise, and hum a tune with no knowledge of any formal musical notation or theory. Rather, the popularity of the Western system is in large part because it has proven easier to use than other alternatives for composition and readability. Western notation has won out over other options through ease of use, not because it is in some sense inextricably tied to what music is.
A secondary choice
We can therefore think of the choice of a notation system to represent semantic information as an aesthetic or pragmatic choice rather than a necessary choice. Faced with some information to communicate, the question of which system of vocabulary and grammar to use depends on a multitude of secondary factors which are not themselves part of the information. For instance, the information “I am lonely” can be communicated as a blunt 3-word sentence, as a song, as a sonnet, or, in the case of a newborn child, as a high pitched howl. In each case, a recipient is able, to a greater or lesser degree to correctly receive and decipher the information, and act accordingly. Aesthetically, we may prefer a sonnet or song to a newborn’s howl, but the effectiveness of both in terms of communication of information cannot be denied.
Fundamentals of language
Since a useful language ultimately just requires the ability to specify one concept as being different to another, we can use any sign or symbol system which enables us to assign recognisable and different signs or symbols to the concepts we want to communicate and / or represent. Provided the hearer has a knowledge of the ‘code’ (i.e. what sign or symbol maps to which concept), communication can occur. When I hear the word ‘lonely’, I know it doesn’t mean a kind of food or a form of exercise or a piece of furniture. Rather, with prior knowledge of English vocabulary, I map the sign onto the concept of being separated from other beings. Likewise, a mother hearing a newborn baby’s howl understands from prior context that this very different sign nonetheless maps onto the same concept: “I am lonely”.
This means that in communicating semantic information, we are ultimately free to choose whatever symbols or signs we like, provided there is a shared consensus of mappings and grammatical rules between writer and reader; speaker and listener. We have seen how colours and sizes of shape can be used to differentiate between symbols, enabling them to represent different concepts (Figure 2). Provided there was consensus over which colour represents which pitch, we could communicate a melody visually using this system. It might not be as visually elegant as Western notation, but as a system for encoding information, it would work.
A question we might then ask is how simple could we go? If we were unconcerned with aesthetic appeal or easy human readability, how much could we simplify down a language system for semantic information and still achieve representation and communication?
We have seen that difference is an important feature of language. We can usefully assign symbols to different concepts provided the symbols are recognisably different. Difference can be provided by using shape, size, colour, position and other parameters. But if we simplify the concept of ‘difference’ to its most fundamental essence, we arrive at ‘yes’ and ‘no’, ‘true’ and ‘false’, ‘on’ and ‘off’, ‘existing’ and ‘not existing’. In other words – a binary choice. Thus we only need two values to abstractly represent the most fundamental form of difference: values we can call “1” and “0”.
This is the basis of binary notation – a system of communication built entirely around the two-digit alphabet of ‘1’ and ‘0’. From this starting point, we can build up a language system ‘from the ground up’, using this most fundamental form of difference to generate a larger set of symbols which have sufficient difference and specificity to represent different specific concepts. In other words, taking the simple two-digit alphabet of ‘1’ and ‘0’, we can build unique ‘words’ of any length we need.
Depending on how large a vocabulary of words we need (i.e. the number of different concepts we want to represent / communicate), we can make our binary words longer or shorter to ensure enough unique variations to encode each different concept.
If the length of a binary ‘word’ is defined as the number of digits (bits) it contains, then a simple mathematical relationship tells us how many different arrangements of 1 and 0 are possible (i.e. how many unique words we can define). For a word length of n digits, the number of possible arrangements is simply 2^n. So if all our words are 2 digits long, we have 4 possible arrangements: four distinct words. If we make our words 4 digits long, we have 16 possible arrangements. Move to 8 digits and we have 256 arrangements. This means 256 different words we can use to denote 256 different concepts: enough for a basic communication system. In fact, conversational English only requires around 1000 words, so using 10-digit binary words would give us an adequate vocabulary for most purposes.
Encoding concepts would be as simple as just mapping each different concept onto a specific binary word. So, for instance, the first word – 0000000000 – might be used to encode the concept of an ‘aardvark. The second word – 1000000000 – might be used to encode the concept of an ‘apple’, and so on.
So long as I share the knowledge of the mappings with another entity – a consensus – we will be able to communicate concepts with each other using nothing but a stream of ones and zeroes. This kind of consensus – programmed by a human mind – underwrites the ability of computers to communicate with one another.
Though binary communication is not particularly intuitive to human beings, this example shows us that we can build up a communication system using only the simplest form of difference: ‘1’ and ‘0’. Thus, again, the concept of difference is seen as being one of the essential components of language, underwriting the ability of a language system to encode concepts distinctly from one another.
Various examples have illustrated that the meaning of what we say – bound up in the information content – is not fundamentally tied to the language we use to communicate. Rather, any language acts as a system of differentiation, enabling unique mappings which can encode concepts using symbols. As the alternative musical notation system illustrated, the choice of symbols is a secondary choice to the actual information we represent with them.
However, what is essential to communication is a shared set of mappings between sender and recipient. There must be some form of consensus among transmitter and receiver; a common understanding that a given mapping encodes a given concept. Moreover, sender and receiver need to share an understanding of how combinations of concepts work: in other words, about the grammatical conventions which enable us to define how a plurality of concepts are interrelated. For instance, if I say “I ate the cat”, these are the rules determining that the “me” in the sentence is dining on the “cat”, and not the other way round.
Interestingly, neither of these factors – vocabulary mappings nor grammar – are inherent in the actual content we communicate with language. The rules themselves are not contained within our speech. Rather, the rules represent a form of shared understanding: a reservoir of meta-information held in common between the minds of transmitter and receiver (Figure 5). This unspoken consensus, a set of rules not itself embedded in our speech or writing, is a prerequisite if information is to be communicated successfully using language: whether in English, Urdu, Western musical notation, binary, or any other.
Figure 5 – communication of information from mind to mind requires representation of information using language, energy and matter (e.g. written English), but also a set of shared rules and context that are not themselves contained in what is communicated.
This issue – of what shared consensus must exist – outside the language and inside the minds of communicators – is key to understanding how information is transferred and used. And, as we will see in the next instalment, it is an issue which strikes at the heart of our inter-relatedness as beings.