Michael Norwitz -- "Carnap's Conventionalism and the Construction of Time in the Aufbau"

Carnap's Der Logische Aufbau der Welt is to date the fullest expression of "methodological solipsism" -- the epistemological doctrine which considers the individual self and its states to be the only legitimate starting point of knowledge. Carnap's starting point is set in the sparsest of desert landscapes; aside from an individual's sensory states, he uses only a single relation to construct our conception of the world. Many have observed that the logic of the Aufbau is not sufficient to the task, and that there are points at which it is necessary to make conventionalist decisions on the direction the construction must take. Here I will examine that thesis in more detail, by re-constructing the world -- showing where and how one can make different conventionalist decisions. The conventionalist choices Carnap makes in the Aufbau are not always obvious; I hope this process of re-construction will make them more obvious.

My intention is not to criticise the Aufbau, as compelling criticisms have come from many sources -- including some of Carnap's later selves -- but to explicate that portion of the construction which deals with time. This explication involves an examination of Carnap's theory of measurement. As a test of the generality of the Aufbau, I will use the tools Carnap supplied to construct an alternate conception of time, a partial rendition of the Hopi as described in Benjamin Whorf's Language, Thought, and Reality (it is contentious whether Whorf's descriptions of Hopi cosmology are accurate, and there are differing views regarding the legitimacy of Whorf's project, but as this paper is not an ethnographic study these are no cause for concern). I have chosen to focus on the construction of time for two reasons: because the Aufbau construction of time is brief, and because time is discussed explicitly by Whorf.

I believe Carnap would welcome such a move, and would think of it as complementing the constructivist project. First, because of the tone of the Aufbau: as pointed out by Michael Friedman, the construction is nothing more than a proposal, a possible way the world might be constructed, with the hope that were the construction carried out successfully we might learn something about the way we actually do construct the world. Carnap is not committed to this particular method being the only or the best way: "This thesis merely asserts the possibility, in general, of a constructional system and especially of a constructional system of the same form as we have used here." Second, because Carnap allows us to direct the construction: as the construct is raised, the logic may take one only so far before certain conventionalist decisions are called for, and the logical construction can begin anew. If we direct the construction using what Whorf would call an SAE ("Standard Average European") conceptual scheme there is no reason why we could not direct it using some other conceptual scheme.

A significant caveat in the pursuit of this project is my choice of interpretation of the nature of Whorf's writing on the Hopi. If I am to construct Hopi metaphysics, then this reconstruction may not be a project Carnap would consider sensical, as he was expressly anti-metaphysical. However, Whorf dismissed claims that the Hopi world-view was excessively metaphysical by asserting that a Hopi would find the Western world-view excessively metaphysical. Presumably to whatever degree the Western world-view could be expressed in a structural account, so could Whorf's presentation of the Hopi. Also, the status of Whorf's project as a metaphysical one (in the non-pejorative sense) is ambiguous. It is clear in "The Relation of Habitual Thought and Behavior to Language" that the purpose of his project in discussing the Hopi is to make clear to us not the nature of our own metaphysics, but places where our use of a certain sort of language leads us to make assumptions about the world.

Finally, I do not wish to appear to be asserting that Hopi metaphysics are somehow facile; my construction of Whorf/Hopi reality is incomplete, and much that I have chosen to leave out exceeded my ability as a constructor. This is another reason for limiting the discussion to the construction of time. The main purpose Whorf's work serves in my paper is to serve as a template for the re-construction, and thus to display the conventionalist elements in Carnap's own construction; hence, it is not necessary for my account of the Hopi conception of time to be accurate (although I do not aim for deliberate inaccuracy). However, it must be noted that Carnap's goal was the construction of the world of science. This explains why he makes some of the conventional decisions he does. You may or may not accept Whorf's claim that Hopi cosmology would be as well suited as our own for the development of a physics.

The construction in the Aufbau uses logic and one relation, the recollection of part similarity (RS). Carnap first constructs a partial temporal ordering of perceived events; in order to construct a complete ordering he must first construct objects which exist abidingly through time. This section on the construction of objects requires for our purposes only a cursory examination, and the explication continues in more depth when we turn to complete ordering. I follow this section by an examination of Carnap's account of measurement of temporal units in Philosophical Foundations of Physics, because it follows naturally from the Aufbau construction, it is of interest in its own right, and further illuminates Carnap's conventionalism, and its provides a broader basis to discuss Hopi cosmology.

I

The first step in our account of the construction is to explain elementary experiences. Rather than taking individual sense data as elementary (for example, "red here now"), Carnap was influenced by the Gestalt psychologists of the time: " ... there is more and more proof that, in perception, the total impression is primary, while sensations and particular feelings, etc., are only the result of abstracting analysis" . Carnap exaggerates Gestalt psychology's collation of elementary experiences into Gestalt units; Edmund Runggaldier observes that Gestalt psychology is more concerned with the mind's tendency to "segregate" or "group" sense data into units. There is every indication that a Gestalt theory of perception was Carnap's inspiration rather than his goal. He idealises for purely constructivist reasons: he is starting from the simplest basis that resembles what is actually given, namely a single unbroken stream of experience. Although these elementary experiences are indivisible we derive from them individuals, both objects and concepts. This is done through "quasi-analysis", which depends on recollection of part similarity.

We shall symbolise the binary relation which Carnap uses to supplement logic as RS, so recollection that elementary experience x is part similar to elementary experience y shall be written xRSy. "Two elementary experiences x and y are called 'part similar' if and only if an experience constituent (e.g. a sensation) a of x and an experience constituent b of y agree, either approximately or completely, in their characteristics ... x and y are recognised as part similar through the comparison of memory images of x and y" . It is possible to read "part similarity" as a technical term, but Carnap does little to unpack it. As Anders Wedberg has pointed out , the use of memory in the RS-relation would, on any standard account of memory, make use of an existent notion of time. Thus any attempt to construct time out of RS will be circular. However, the Aufbau is a rational reconstruction of the way the physical world might be generated, not a psychological account of the way an individual, call him A, does generate the physical world. RS is eliminable, and can be defined in a way that does not presuppose time. One possible definition of RS, for example, would be "the founded [very roughly, entrenched] relation extension from which we can construct in a given way a sufficiently high level object, still to be chosen, which shows certain empirical characteristics." Formulated in this manner (ostensively rather than intensionally), RS could be used to generate the notion of time without presupposing it.

RS is also a quasi-analytic notion, which Carnap develops to suit his derived version of Gestalt psychology. "Proper analysis" consists of distinguishing parts, or properties, or characteristics, of a complex object. Applied to a class or set, proper analysis involves discerning its members. "Quasi-analysis" is the converse procedure, and involves discerning (some of) the sets of which a basic element is a member. From a collection of elementary experiences, each of which stand in the RS-relation to the other, we extrapolate a common quasi-constituent. This constituent is not a part of an elementary experience, because they have no parts. It is a logical construct that we generate using RS. Imagine a series of elementary experiences, which (using physical language) include a perception of a red pen, of spilled blood, of an apple. There will be many similarities between these experiences which will be of varying levels of intuitive obviousness; colour similarities tend to be fairly obvious. Noting that these experiences are all similar by recognising part similarity allows us to extrapolate the common redness of them, and from the set of experiences which are grouped together in this way we construct the property of redness.

II

We now have enough technical vocabulary to construct the "partial time order", on the way to constructing the "complete time order".

Carnap defines the RS-relation as asymmetrical; given two immediate elementary experiences x and y, recollection that x is part similar to y requires a comparison between x and a (perfectly remembered) image of y; hence xRSy would not be compatible with yRSx. Each remembered elementary experience was an immediate elementary experience at some point, and at that point of "immediacy" it would have been compared to the stock of remembered images, and thus in a way sorted by RS. The RS-relation does not require that either of its relata be immediate in the sense of occuring now. Once made, however, comparisons are permanent. Within the mind of the rational reconstructor Ar, who may be thought of as the protagonist of the Aufbau, we consider the memory image of x as epistemically equivalent to the image of x ; both x and y can be remembered. As images do not fade with time we may (perhaps) assume that the memory of the comparison does not fade with time. So for any two elementary experiences x and y, if xRSy than x is later than y. We see that a temporal ordering is relatively easy to derive from RS. Because Carnap defines this relation as transitive we may also say that one elementary experience is earlier than another if an "RS-chain" exists from one to the other. This is only a partial ordering (what Carnap calls a "preliminary ordering") because it is not without gaps; we can not decide for every pair of elementary experiences which of these is temporally prior. It is possible that there may be collections of elementary experiences, each of which are sorted by RS into a temporal ordering, but which have no common parts. In order to bridge the gaps between discrete temporal sequences we need the concept of something which exists abidingly through time. According to Carnap, a complete ordering requires regularity of physical processes, and that requires construction of physical objects: "... the time ordering of experiences which is based on 'time perception' is incomplete and becomes a completely ordered sequence only through inferences on the basis of known psychological and especially physical relations." I think what is being implied here, is that if an object exists permanently, or at least abidingly, then there is a space-time for it to exist permanently in.

The simplest assumption to make is that there is a single space-time. Hence it would be natural to imbed previously disjoint temporal sets within a single continuum. This is of course also a conventional assumption, made to direct the construction, because Carnap is trying to construct a world compatible with physics.

Prior to the construction of physical objects the Aufbau ceases to be constructional and becomes programmatical. A mere construction of what Carnap calls the "colour solid" (a three-dimensional representation of the colour scheme which is used to map our perceptions of similarities between colours), which is a preliminary step to the construction of objects, is immensely complicated. To go beyond that tries the patience (if not the ability) of even the most adept and dedicated philosopher/logician -- though it is not entirely clear whether Carnap intended the programme to be one day carried out by a successor (or a school of successors), or left to the reader's imagination what completion of the programme might be like.

Carnap finds it practical to first construct the space-time world and then to commence filling it with objects. Dealing only with the visual sense, he assigns colours to the various world-points which comprise the four-dimensional world. Why four dimensions? The decision is arrived at for conventional, pragmatic reasons: From an empirical assumption about the two-dimensionality of the visual field, it follows that n (the number of dimensions) _ 3. From the (in realistic language) disappearance and reappearance of things in the visual field, it follows that n _ 4; hence, the dimension number of space is at least 3. Hence, we fix the dimension number of the order of world points at 4, that of space at 3. So our assignments to our world are of the form [x,y,z,t], where t = time.

Our protagonist Ar has a point of view, that is, a continuous curve of world-points having as their focus a particular point in the interior of the head; Carnap also makes the physical assumption (to simplify the eventual physical system, though it is odd to include it at this stage of the game) that light travels in a straight line from the objects in the point of view to the eye, and that the optical medium between the eye and the seen things remains homogenous over time. Note we are presupposing a Euclidean space-time. In addition to being most conducive to a simple construction, this conventional assumption about the physical world also expresses Carnap's judgement under what conditions observation would be maximally or most reliably veridical. Having abstracted colour terms through quasi analysis, Ar can assign an appropriate colour to each world-point of the corresponding point of view; on the assumption that the construction is leading to a true rendition of the world, this would entitle us to say that the point which Ar sees corresponds to a point on the outside world of the same colour (so when there is a red spot on my visual field then there actually is, or I am entitled to say there actually is, a red object in front of me) .

Ar assigns colours to unseen world points in such a way that each belongs to a world line: a continuous curve or curve segment in which each world point is associated with each value of a time coordinate within a given interval; in physical language, a point in the outside world once seen shall be assumed to have existed previously and to continue to exist after the present time t, again, within a given interval, and this existence is continuous. Ar also assumes that once a world point has been assigned a colour, it retains that colour across the world line unless there is a reason to believe otherwise; in physical language, objects which are within the world as constructed retain (or are assumed to maintain) their colour and shape unless reasons are had to believe they have changed. This is also a simplifying, conventionalist assumption.

Finally, we are allowed the regularity of physical processes, through a conventional completion of the physical world "by analogy": presented with two discrete space-time regions whose assignment of sense qualities are extremely similar, if one of the regions includes assignment to points which are lacking in the corresponding region(s) in the other region, Ar assigns analogous points in the latter region(s). There are other world points to which Ar assigns colours even though they are not in his point of view, and which are connected into a two-dimensional or three-dimensional area; these become, in constructional terms, the surfaces of bodies. At this point Ar has constructed a model of a physical object, and can now approach the question of something existing abidingly through time, which will lead to complete time order. The analogy applies both to the spatial and the temporal parameters: if a process is unobserved for a period of time, but a temporally significant portion of that which is observed is similar to another process which has been observed in full, then Ar assumes that during the period in which no observations were made, the former process behaves in a way analogously with the latter; likewise, if a spatial part of an object which was previously perceived in full remains unobserved, while the rest of the object is perceived again, Ar assumes that the unobserved spatial part is analogous to the corresponding part(s) of the originally perceived thing. Both of these assumptions are made if there are no overriding reasons for thinking that they ought not to be applied.

Having established physical objects and processes, Ar can construct a complete time order, which unlike the partial time order is connected, that is, a relation defined on that order (say, earlier-than) or its converse will always hold. In a connected time order there are no gaps, no discrete segments of temporally ordered events; for every pair of elementary experiences we can decide which of them is temporally earlier.

III

Carnap does not discuss at length questions of measurement in the Aufbau, though he does in later writings, notably his collection of lecture notes Philosophical Foundations of Physics, in which he is explicitly conventionalist. As is a continuation of a common theme in his work, we will use the later text to complete a Carnapian account of the construction of time.

Measurement of time intervals involves considerable problems. Unlike spatial intervals they can not be easily manipulated (or at least, there is no obvious temporal analogue to the use of a solid body to represent a spatial interval). The most elementary operation Ar can carry out on time intervals -- i.e. world lines with demarcated end-points in the temporal parameter -- is additivity: "Suppose we have one event a that ran from time point A to time point B and a second event b that ran from time point B to time point C... . The initial point of b is the same as the terminal point of a, so the two events are adjacent in time ... . The length of time from point A to point C can now be regarded as the result of combining a and b, not in the physical way lengths are combined, but in a conceptual way, that is, by the way we look at the situation . Carnap introduces a conceptual operation "¡" and a rule of additivity for the measurement of temporal length T: T(a¡b) = T(a) + T(b).

Ar does not only add together events which are adjacent in time (otherwise, this would make calculating his weekly paycheque, among other things, rather difficult); this requires a joining operation, which itself requires him to define a unit and a rule of equality over time segments. He can derive these from the idea of a periodic process. This is a process which recurs again and again and again: a heartbeat, a swing of a pendulum, and the rising of the tides, are all periodic. Ar then affirms the applicability of the additivity operation that the length of n number of consecutive intervals is the sum of n intervals.

For a rule which will define a unit, Ar need simply dictate that the duration of any one period of a process is a unit. The rule of equality then is: two events are of equal length of time if they both contain the same number of units of the periodic process. It is an elementary matter to add together two discrete and nonadjacent events by determining the length of each through the measure of the basic unit. The act of choosing which periodic process will define one's time unit -- whether to prefer the periodic vibration of the cesium atom over the heartbeat of the Queen or the occurrence of a lunar eclipse -- is a conventional decision, dictated by the desire for a simple world system. Carnap in neither of these texts produces criteria of simplicity. There are disadvantages which may for pragmatic reasons be avoided by not basing a unit on an individual's heartbeat (which would have the effect of all other processes in the world slowing down when that individual is asleep and speeding up when the individual is awake -- which would be the case even for an individual with as relatively sedate a lifestyle as the Queen) or by not basing a unit on the occurrence of a lunar eclipse (which would make too many of our ordinary operations on temporal processes require unwieldy fractional equations).

IV

Benjamin Whorf, in Language, Thought, and Reality, recounts the linguistic categories of the Hopi, gleaned from years of study on the Hopi reservation with a native interpreter. On the basis of his account, Hopi has no words which translate directly into "time" or SAE temporal concepts (by which Whorf means Euclidean concepts).

As SAE categorises phenomena in terms of past, present, and future, so Hopi categorises phenomena in terms of manifested (or objective) and unmanifest (or subjective). The former comprises events which are or have been accessible to the senses, with no distinction between the two. The latter comprises events in the future, and in addition mental phenomena such as fears, beliefs, and desires; properly speaking (though still crudely) this might be spoken of as comprising dispositions (or Aristotelian "natures"), as it also includes that which is "in the heart" (as the Hopi prefer to say) of animals, plants, objects, and in its widest range in the heart of the Cosmos itself. Although there is a borderline, a manifesting, between the unmanifest and the manifest, which may be thought of as SAE's "present", most of what we call the present belongs to the objective realm. Their borderline state, which represents the point of becoming manifest, suffers from a sort of conceptual vagueness in Whorf's presentation which might be thought of as analogous to the "disappearing present" discussed by some philosophers of time. The unmanifest is not a temporal future; there is nothing in the subjective state which corresponds to the sequences and changing physical configurations which are found in the objective state.

The subjective also can not be used to discuss spatial relations, and the objective discusses them in a purely operational sense. The operational meaning of an expression is given by a rule which relates the expression to some concrete process or event. Both spatial and temporal magnitudes are a function of the complexity of the operations connecting events (in fact, the two are interrelated in Hopi):

Two events in the past occurred a long "time" apart (the Hopi language has no word quite equivalent to our "time") when many periodic physical motions have occurred between them in such a way as to traverse much distance or accumulate magnitude of physical display in other ways. The Hopi metaphysics does not raise the question whether the things in a different village exist at the same moment as things in one's own village, for it is frankly pragmatic on this score and says that any "events" in the distant village can be compared to any events in one's own village only by an interval of magnitude that has both time and space forms in it. ... Both the "here" happening and the "there" happening are in the objective, corresponding in general to our past, but the "there" happening is the more objectively distant, meaning, from our standpoint, that it is further away in the past just as it is further away from us in space than the "here" happening.

The objective contains all extensional aspects of existence. Yet as attributes of that extension become farther removed, it merges again with the subjective. Events taking place in the legendary or pre-legendary past are as much subjective as objective. The equivalent in SAE terms would be macrocosmic events such as the pulsing of a quasar, microscopic events among electrons and quarks, and primeval events such as the Big Bang; all at a distance of length, size, or time which makes them operationally "distant" in a way which the shop down the road, the breadbox in the kitchen, and one's last meal are not. Of course the distances in these different parameters are along a continuum, but it is still a real distance.

Turning again to a discussion of measurement, we find that according to Whorf, SAE languages apply two sorts of pluralisation: to "perceptible spatial aggregates" and to what he calls "metaphorical aggregates" . The former applies to situations such as "ten men", which involve an objectively perceived group. A group such as "ten days" is a mental construct out of a concept of "cyclicity", which involves imaginary plurals. It captures a sense of "becoming later and later", as does the Hopi concept, though in a different way. While the SAE languages number days using cardinals, the Hopi reserve cardinals for entities which can form an objective group, and pluralise temporal units through ordinals. Consequently the Hopi do not pluralise time units themselves; to use Whorf's example rather than saying "they stayed ten days" the Hopi would say "they stayed until the eleventh day" or "they left after the tenth day". The measure of a period of time becomes a relation between two elements in lateness. This is displayed in the Hopi metaphysics which sees successive days as successive appearances of the same day, which can be affected by one's actions (just as successive appearances of the same friend at your house can be affected by your actions towards them on any particular visit -- if you set their hair on fire on Monday, they will appear a little singed on Thursday).

Unlike SAE languages, Hopi does not designate phases of temporal cycles using nouns. SAE marks off a certain portion of, for example, the seasonal cycle, as "the summer". The closest analogue in Hopi is a particular type of adverb (distinct from the other Hopi "adverbs"), which would translate on the order of "when it is summer" or "while summer-phase is occurring". There is no abstract object, "the summer", merely a characterisation of when it is hot. In some ways Hopi does not suffer from some of the ambiguities which have given rise to confusion (and much metaphysical speculation) in the SAE languages. The typing of phase terms as adverbs suggests nothing about time except the perpetual passing of it. Another example is the present tense which also denotes nomic statements, and a future tense which also seems to represent the earlier (anticipation), and the later (afterwards, what will be). Rather than distinguishing past-present-future plus timeless (nomic) truths, the Hopi distinguish between reportage (answering to our past and present -- the manifest, objective realm), anticipation (answering to our future -- the unmanifest, subjective realm), and making of nomic statements (answering to our nomic present).

We can actually generalise the earlier statements about phase cycles, and observe that the Hopi seem to express reality through verbs, as the SAE languages do through nouns. In the objective realm facts are expressed in terms of colours, outlines, movements, and other sensory statements; in the subjective realm they are expressed in terms of a myriad of dispositions, which are preparation for eventual manifestation. There are whole classes of things which the Hopi do not acknowledge as objects at all: "In the Hopi language, 'lightning, wave, flame, meteor, puff of smoke, pulsation' are verbs -- events of necessarily brief duration cannot be anything but verbs. 'Cloud' and 'storm' are at about the lower limit of duration for nouns. Hopi ... actually has a classification of events (or linguistic isolates) by duration type, something strange to our mode of thought" . In SAE we may consider, say, a "fist" to be an object, while we wouldn't say someone is "fisting".

V

That represents the bulk of Whorf's account of the Hopi concept of time; we now turn to attempt its "construction" in the manner of Carnap. A new construction will require a new protagonist, so I will call this one Ac . There is little problem with partial time, which derives directly from the RS-relation and which involves minimal metaphysical claim; greater difficulty will come in the construction of completed time (which involves physical objects and processes) and measurement.

Suppose on the SAE model Ar constructs a four-dimensional space-time which he commences to fill with objects. The past and the future are represented by a line extended in the positive and negative directions, with the positive values stretching back into the past and the negative ones into the future, and the present time set at zero. The value attribution at each point in the line is not permanent of course, but relative to where one is along the line. Although it is assumed that the enumerative system in Philosophical Foundations of Physics is based on cardinal numbers, Ac in contrast will be basing her system on ordinals (although my inversion of Ar's time-line might seem unintuitive, it is algebraically trivial, and has the advantage of being more similar to Ac's formulation, so that Ac does not pluralise manifest events with anything like negative ordinals).

Ac does not acknowledge an objective past and future with sequences and successions within which are mapped distances and changing physical configurations. The ordinals (which represent multiple occurrences of the same event, which can be altered or affected by one's actions) can then be characterised with one of her temporal adjectives, such as "while summer-phase is occurring". She might represent space with three dimensions similar to Ar's. However there will be no "points" as such, rather each position in the visual field will be a point-vector in the negative direction, which represents the concept that her present is a point of inception, of becoming manifest, rather than a sort of cut between two lines stretching off in either direction (point-vectors being a somewhat "degenerate" concept, an alternate approach would be to incorporate into this stage of the construction some function Ä defined on all ordered quadruplets [x, y, z, t] in which x, y, z represent spatial parameters and t represents time, so that Ä[x, y, z, t] = [x, y, z, t - 1]; this function to represent the yet-to-manifest nature of whatever is specified in that space-time point However, it has the disadvantage that it suggests there is a discrete point [x, y, z, t] to which the function Ä can be applied, and the Hopi do not recognise discrete temporal states).

Ac has a "point of view" and assigns colour-values to the points (or point-vectors) in that field. She may construct physical objects and processes in a manner not dissimilar to the manner of the Aufbau, however the constructions will be interpreted differently to fit the alternate space-time. Although analogy will still be used, there is no assumption of objects existing in a steady state unless reason is given to the contrary; rather, objects will carry in their hearts their future states and analogy is used to prepare for what those states will be when manifested. Properly speaking a collection of points (or point-vectors) will not necessarily qualify as an "object" at all, merely a potential object, depending upon the equipollence of its temporal extension with a certain ordinal. Ac will have to set a convention for the order of qualification, with "cloud" and "storm" just qualifying by that order. If its temporal extension is below a certain order it will qualify as an event, be it "lightning, wave, flame, meteor, puff of smoke, pulsation". This is important because the construction of objects is a preliminary requirement for the construction of completed time in the Aufbau.

Of course Carnap's instructions for how to devise a system for measuring time would not be suitable for Ac's project. While Carnap decides conventionally on a periodic process to use as a temporal unit, each cyclical process Ac examines will be operationally defined, and it is not obvious that every ordinal will be representable by a phase series of which she will be able to construct a correspondence between it and any other phase. Ac would need to be constantly aware that the measure she uses is relative, perhaps through the use of a subscript, whereas the constructor in the SAE system can easily forget this fact and treat the basic unit as absolute, once it has been decided on.

Ac will use ordinal rather than cardinal addition. The rule of additivity poses more problems, as T(a¡*b) would only be defined for an a and a b in the subjective realm. We might still be able to add together certain well-established unmanifest cycles by analogy, but that would be an instrumental decision; the temporal objects whose lengths are being added don't "really" exist. Beyond that consideration, the rule of additivity might thus be applied in much the same manner. Even on the point-vector analysis Ac is not required to enter into the realms of vector addition, as it is the lines along which the vector's bases lie which are being added; hence none of this forces a decision between the point-vector and the Ä-function interpretations of the Hopi space-time.

VI

Despite the logistic problems in the Aufbau, as a constructional system it is sufficiently flexible, I hope I have indicated, to act as a foundation for alternate ways of seeing the world. I hope also I have not done additional violence to the Hopi cosmology in the construction via Ac than I did in my explication of Whorf; while this is not an ethnographic research project, that much consistency ought to be maintained. However, it matters little. Whorf's compressing of all Western world-views into a "standard" "average" "European" (not to mention Euclidean!) world-view is inaccurate and oversimplistic. Carnap's (also Euclidean) construction operates through our accepting certain levels of analogy, both explicitly in the construction of physical objects and completed time, and implicitly in the project of mapping our phenomenal world-view onto a four-dimensional geometric structure (this last is an analogy accepted in the sciences). Of course, a reconstruction is not an analogy; but note that because of its simplifying assumptions (for example, Ar having perfect memory) what he is constructing is not the SAE point of view -- which would require decades of cognitive science research to actually describe accurately -- but something analogous to it, an idealised version of it. Consequently what is required is a different set of analogies which would supply a different set of conventional decisions at crucial points in the construction, and Whorf's account of Hopi cosmology does a sufficient job of providing these.

For more information online, cf. Rudolf Carnap Related Web Links

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