Abstraction

APR 15, 1985

© Copyright 1985 by Ralph E. Kenyon, Jr.

Abstraction. (1) In traditional logic, the process of deriving a universal from particulars. . . .¹

Abstract:  In this paper I will trace the evolution of a particular sense of the term 'abstraction', define terms necessary to give a precise technical definition for this sense as it has evolved, and illustrate application of the term in that technical sense.

William of Ockham (1285-1349) differentiated between intuitive cognition and abstractive cognition.

Intuitive cognition is defined as an act of apprehension in virtue of which the intellect can evidently judge that the apprehended object exists or does not exist, or that it has or does not have some particular quality or other condition; in short, an intuitive cognition is an act of immediate awareness in virtue of which an evident judgment of contingent fact can be made.

Abstractive cognition is defined as any act of cognition in virtue of which it cannot be evidently known whether the apprehended object exists or does not exist, and in virtue of which an evident contingent judgment cannot be made.²

In this early usage the element of "evident" uncertainty is what distinguishes between abstractive and intuitive cognition.  The lack of immediate evidence suggests that the uncertainty comes from some absence of data.  It is the loss or absence of data which will figure most prominently in the future evolution of this term.

Tommaso Campanella (1568-1639) adapted the distinction to the process of obtaining knowledge.

Knowledge of the external world can be obtained either by intuition or by abstraction. . . . By abstraction, one obtains only an indistinct and confused image of a thing.³

Campanella moves more firmly in the direction of losing data with his characterization that the knowledge is indistinct, and he is focusing on the process of obtaining knowledge, as opposed to "cognition".  The Cartesians [Descartes 1596-1701] moved to focus on the act in which knowledge by abstraction is obtained.

Philosophers working within a Cartesian tradition accept a "simple act" theory.  On this traditional view, we possess a faculty called abstraction that, for example, peels the whiteness of white objects from these objects and holds the whiteness up before our mental eye; once we have whiteness clearly in focus, we can label it with the term "white" and thus can acquire knowledge of how to use this term.  This act of abstraction, like the act of intuiting that p, is specifically mental, simple, and unanalyzable.⁴

Even while characterizing the act as unanalyzable, the example given suggests an analysis which is in agreement with the loss of data suggested by earlier usages.  In dealing with the relationship between general terms and particular terms, Locke (1632-1704) moved away from the notion that the act of abstraction is unanalyzable.

He expresses the problem by asking, "Since all things that exist are only particulars, how come we by general terms; or where find we those general natures they are supposed to stand for?" His answer is that words are general by being signs of general ideas, and we form general ideas by abstraction, "separating from them the circumstances of time and place, and any other ideas that may determine them to this or that particular existence".⁵

Locke has explicitly focused on this loss of data and has described it as the method of abstraction.

I shall propose a precise technical definition for 'abstraction' once some preliminary definitions are in place.  But as motivation for what follows, let me just say here that abstraction is an act, or the process entailed in that act, in which particular details are "left out", or the result or output of that process.  My technical definition will be in agreement with this informal characterization.

I will define some preliminary technical terms, using ordinary language, and then define 'abstraction' in a precise way.  But first, I need several preliminary notions. 

'Structure' is an undefined term; states, devices, messages, media, etc. are all structures.  'Information' is a technical term as characterized by modern information theory.⁶

Information is measured in bits.  The amount of information in the position of a simple switch (on or off) is one bit.  Information flow is measured in baud, which is the digital equivalent a frequency.  One baud is equal to one bit per second.  A rate of 10 baud or bits per second is the same as a frequency of 10 Hertz (1 Hz. = 1 cycle per second).  Engineers use 'frequency' in reference to sinusoidal signals (analog) and 'baud-rates' in reference to square-wave signals (digital).  The distinction is unimportant for my purposes, but it is customary to use one or the other terms in different contexts. 

A device is any structure having inputs and/or outputs.  A simple device has only one input and/or one output.  A complex device has at least two inputs and/or outputs.  Complex devices are made up of simple devices.  A simple device having only one input is a sink.  A simple device having only one output is a source. Information is generated at a source, and consumed at a sink. 

A medium is any structure capable of carrying information, and which may act as a sink and/or a source.  One characteristic of a medium is its band-width; the band-width is the range of frequencies it will carry, pass, conduct, etc.  The band-width of a medium is a measure of it's information carrying capacity.  A telephone circuit is designed with a band-width of 2600 Hz; it can carry information at the rate of 2600 baud.  A medium is a device, but there are devices which are not media, for example a sink or a source. 

Homogenous media may be a source or sink at any point.  Examples of homogenous media include light, sound, water, etc.  Media which are not homogeneous are Heterogeneous. Examples of heterogeneous media include vision, radio, print, etc.  The department of Media Ecology at New York University characterizes media in a very general sense as any communication situation.  I think my definition here captures that generality.  One special medium is the environment.  The environment is a source and sink for every device. 

A transformer is any device which inputs and outputs to media of the same kind.  Examples include an electrical power transformer, a set of linear transformation equations (numbers in, numbers out), a computer (text in, text out), a mirror (light in, light out), and etc.  A device which is a transformer, may also be a transformer in the other direction, but this is not always the case. 

A transducer is any device which inputs and outputs to media of different kinds.  Examples include an audio speaker (electric alternating current in, audio frequency sound out [microphones go the other way]), a numerical string conversion function in a computer language (strings in, numbers out [or vice versa]), a computer (tactile key-strokes in, video display terminal out), a piezoelectric crystal (mechanical pressure in, electric voltage change out), a rod cell in the retina (light in, nervous impulses out), etc.  A device which is a transducer, may also be a transducer in the other direction, but this is not always the case. 

A filter is any transformer whose output range of frequencies is a subset of its input range of frequencies.  A filter generalizes its input. 

A noise generator is any transformer whose output range of frequencies is a superset of its input range of frequencies.  A noise generator particularizes its input.

Most physical devices may be subject to the entire range of frequencies capable of being carried by the input medium, but do not respond to all these frequencies; they effectively act as filters in this regard. 

Transformation is the process of information "going thru" a transformer; transduction is the process of information going thru a transducer. 

Abstraction is a series of one or more transformations and/or transductions connected in series (input of one device connect to the output of another device).  Since devices may have more than one inputs and outputs, this allows for complex parallel connections in devices which instantiate abstraction.  Most abstraction includes filtration and noise generation.  An abstraction is the output of abstraction.

This definition differs from the evolved use by the inclusion of noise generation.  The traditional use would be accommodated in abstraction accomplished by devices which included no noise generators.  There are compelling reasons (which will become evident later) not to introduce the grotesque asymmetry which would be produced by such a restriction.  However, there are anomalies in the use of the term abstract.  A message is said to be "more abstract" than another if it can be produced from the first by a device which accomplishes more filtration than noise generation. 

Special devices include a translator, which transforms one coding to another if the medium is viewed as the symbols, or transduces one language to another if the media are viewed as different languages.

A level of abstraction is the value of a function from inputs and outputs in a device to integers evaluated at inputs and outputs; the function must satisfy the following constraints: the value at the output of a filter is always greater than the value at its input, the value of the output of a noise-generator is always less than the value at its input, and the function has only one value at points where the input of one device is connected to the output of another device (it has only one value at a homogenous medium).  The function which produces levels of abstractions maps the of flow of information in a complex device. 

An encoding is a relation from a set of possible states to binary digits. An encoding is unambiguous if it is a function.  Encoding is the process of selecting a value (binary digits) for a possible state (using an encoding). 

A decoding is a relation from a set binary digits to a set of possible states.  A decoding is noiseless or noise free if it is a function. Decoding is the process of selecting a possible state for a value (binary digits) (using a decoding). 

An encoding or decoding is unequivocal if it is one-to-one.

A device receives a message at its input, decodes it into possible internal states, undergoes a state transformation, encodes its internal state into a message and transmits the new message at its output.  The state transformation may be the identity transformation, and the output message may be delayed (including permanently, that is, for some internal states, no message is transmitted, for others, a message is transmitted after a delay, etc).

In the decoding of a noisy message, there will be a number of possible assignments which may be made in selecting the state to associate with the message.  Selecting a particular state is called particularizing. 

Consider the following cases. 

1. A state is unambiguously encoded and transmitted by a source; it is processed by a complex medium which makes no changes; the message is decoded into the state corresponding to the state encoded by the source. 
2. A state is unambiguously encoded and transmitted by a source; it is processed by a complex medium which filters certain parts, and injects noise in their place; the noisy message is decoded (fortuitously) into the state correctly corresponding to the state encoded by the source. 
3. A state is equivocally encoded and transmitted by a source; it is processed by a complex medium which makes no changes; the message is decoded into the state corresponding to the filtered state encoded by the source.  (The message sent is a generalization, and is properly received)
4. A state is equivocally encoded and transmitted by a source; it is processed by a complex medium which injects noise in the places filtered; the noisy message is decoded (fortuitously) into the state corresponding to the state filtered by the source.  (The message sent is a generalization, and is improperly received, in that the decoded message corresponds not to the generalization, but the state whence came the generalization.)
5. A state is equivocally encoded and transmitted by a source; it is processed by a complex medium which makes no changes; message is decoded into the state corresponding to the state encoded by the source.  (The message sent is a generalization, and is properly received, in that the decoded message corresponds to the generalization.) The receiver undergoes a state transformation (fortuitously) to the state originally filtered by the sender.

These cases illustrates various information processing situations.  In each case the final state is an abstraction from the state encoded. 

We are now in a position to interpret the definition of 'more abstract'.  A message is said to be more abstract than another if it can be produced from the first by a device which accomplishes more generalization than particularization.

Abstraction is composed of two processes, transformation and transduction, which may be affected by loss (generalization) or gain (particularization).  Rules of inference which could mimic abstraction include four rules: transduce (change media), transform (change "shape" in the same medium), generalize (drop something), and particularize (add something).

Analogy.  Reasoning by analogy is "explained" by two abstractions, generalization and particularization; a structure is generalized to obtain a second, more abstract, structure and that more abstract structure is particularized to form the structure "analogous" to the first.

Concept Formation. In decoding, a message received is used to select from a number of states.  If there is a predisposition to select from a particular subset of all possible states, (the decoding is not onto) then the decoding takes place into a pre-existing structure (the range of the decoding).

Concept formation is the addition of possible states to the pre-existing structure, usually by adding two states which can both be generalized into a state already included in the pre-existing structure.

In set notation, the decoding assigns elements of the power set of some set of states to the incoming message.  Concept formation can only occur if the decoding function is not one-to-one, and occurs by "learning" to differentiate incoming messages, which were previously decoded to one subset, to different subsets, preferably disjoint, whose union is the original subset.  One of these subsets becomes the "concept", the other becomes what it is not.

We may view human beings as complex devices which input with their senses and output words.  There are several significant stages in the process, but abstraction applies to the overall process as well as to individual stages.

By the fact that our senses are sensitive to a limited range of frequencies, filtration occurs at the first stage.  Our senses abstract a limited range of frequencies from the environment.  By the fact that individual receptor cells absorb only a few characteristic bits of information from this range, further abstraction occurs.  This abstraction is also a generalization into pre-existing neurological structures or categories.  From the information transmitted by thousands of such cells in parallel, higher level structures respond to certain patterns of individual stimulation.  This response is a selection from among the many possible patterns, and so is a further stage of abstraction; it is still primarily generalization as well.  From these patterns, we particularize by selecting from among many verbal associations (ambiguous decoding) a word or words to represent the pattern of frequencies in the environment.  The "correctness" of the particularization derives from continuous experience with many messages over a long learning period, but in the individual selection of words, the final stages of the process matches simple noise-generation, albeit non-random.  It is inputs from other devices, including memory (devices whose outputs are delayed) which select the decode function to use for the individual time.  (A word by itself is ambiguous; in a sentence it is less ambiguous; in a paragraph even less so.)

Motivation for allowing particularizing as included in the mechanism of "leaving out" is that we can consider a more abstract datum as an ambiguous equivocation from all possible data which could be generalized to the abstract datum.  If we consider the abstract datum as somehow including the function which performs the equivocation in forming the general datum along with all the possible data which form the domain of the function, then particularizing becomes selecting one of these possible data and discarding the rest.  We still have picking and discarding as the basis of particularizing.

Some discussion on this is available at "more on abstracting".

References & Notes

1. Boruch A. Brody, Logical Terms, Glossary of, Encyclopedia of Philosophy, Macmillan, New York, 1967, Vol. 5, p. 57.

2. Ernest A. Moody, William of Ockham, Encyclopedia of Philosophy, Macmillan, New York, 1967, Vol. 8, p. 308.

3. Bernardine M. Bonansea, Campanella, Tommaso, Encyclopedia of Philosophy, Macmillan, New York, 1967, Vol. 2, p. 12.

4. Richard Rorty, Intuitionism, Encyclopedia of Philosophy, Macmillan, New York, 1967, Vol. 2, p. 208.

5. D. W. Hamlyn, Epistemology, History of, Encyclopedia of Philosophy, Macmillan, New York, 1967, Vol. 3, p. 23.

6. Fred I. Dretske, Knowledge & the Flow of Information, MIT Press, Cambridge, 1981.