Temporal Orientation
//Bead Panel// from a baby carrier, Bahau people, Borneo 20th century, 30 x 29 cm. From the Teo Family collection, Kuching. Photograph by D Dunlop.
Bead Panel from a baby carrier, Bahau people, Borneo 20th century, 30 x 29 cm. From the Teo Family collection, Kuching. Photograph by D Dunlop.

Here is a thought experiment to articulate a relationship between the order of events in some chain, and any thermal sensations that are part of those events. The result is reported as the temporal orientation of the chain. Let

\mbox{\fontsize{14}{18}\selectfont $ \Psi ^{\sf{P}} = \left( \sf{P}_{1} , \sf{P}_{2} , \sf{P}_{3} \ldots \right) $}

be an ordered set of events where the subscript event index is arranged just like counting, in the usual numerical order. And let each event be characterized by its temperature T. The temporal orientation is a binary description of how these temperatures vary with the event index. Consider a pair of events Pj and Pk arbitrarily selected from the chain Ψ. Characterize the pair by the difference in their order indices

\mbox{\fontsize{14}{18}\selectfont $ \Delta_{i} = k - j $}

and their temperature difference

\mbox{\fontsize{14}{18}\selectfont $ \Delta T_{i} = T_{k} - T_{j} $}

Next consider a large number N of event-pairs sampled from Ψ. Calculate the average order-difference in the sample

\mbox{\fontsize{14}{18}\selectfont $ \widetilde{\Delta} = \frac{1}{N} \sum_{i=1}^{N} \Delta_{i} $}

and the average difference of temperature in the sample

\mbox{\fontsize{14}{18}\selectfont $ \widetilde{\Delta}T = \frac{1}{N} \sum_{i=1}^{N} \Delta T_{i} $}

Then characterize Ψ by finding the way they vary with each other

\mbox{\fontsize{14}{18}\selectfont $ \chi = \frac{1}{N} \sum_{i=1}^{N} \left( \Delta T_{i} - \widetilde{\Delta}T \right) \left( \Delta_{i} - \widetilde{\Delta} \right) $}

The result is reported as a binary description of the relationship

(1)
\acute{\varepsilon} \equiv \begin{cases} +1 &\text{ \sf{if} $ \chi \ge 0 $ } \ -1 &\text{ \sf{if} $ \chi < 0 $ } \end{cases}

Definition: The number is called the temporal orientation of Ψ.
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Sensory interpretation: if the temporal orientation is positive, then usually perceptions of temperature would increase if events were felt in the same order as numbered. Usually but not always; temperature changes would not necessarily be monotonic, just correlated with numerical order. Alternatively, if the temporal orientation is negative, then P would typically be cooling if events were experienced in the same order as numbered. Roughly speaking; if Ψ is compared to a movie, then indicates whether the movie shows events to be warming or cooling. Please note that this is not quite the same as saying that P is truly warming or cooling because the movie could be running backwards. Temporal orientation describes a relationship between temperature and order, not temperature and time.
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Next step: historical order.
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