Speed-Accuracy Trade-Offs (in Physical Cognition)

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Physical Cognition: Impetus and Representational Momentum

Non-experts reliably judge that a projectile exiting a spiral tube will subsequently follow a spiral trajectory (McCloskey, Caramazza, & Green, 1980). Why?

Sometimes when adult humans observe a moving object that disappears, they will misremember the location of its disappearance in way that reflects its momentum; this effect is called representational momentum (Freyd & Finke, 1984; Hubbard, 2010).

The trajectories implied by representational momentum reveal that the effect reflects impetus mechanics rather than Newtonian principles (Freyd & Jones, 1994; Kozhevnikov & Hegarty, 2001; Hubbard, Blessum, & Ruppel, 2001; Hubbard, 2013). And these trajectories are independent of subjects’ scientific knowledge (Freyd & Jones, 1994; Kozhevnikov & Hegarty, 2001). Representational momentum therefore reflects judgement-independent expectations about objects’ movements which track momentum in accordance with a principle of impetus.¹

We might therefore conjecture that fast processes explain the spiral trajectory judgements observed by (McCloskey et al., 1980).

How Do Fast Process Influence Explicit Verbal Judgements?

In the case of physics, this appears to involve fast processes influencing the overall phenonemenal character of experiences, reflection on which which in turn influences judgements.

Phenomenology connects fast and slow processes indirectly. That is, there are content-respecting relations between fast and slow processes which may not require inferrential connections.

Phenomenology connects fast and slow processes leaving room for discretion. That is, individuals are free to make judgments which conflict with model implicit in the fast processes, as expert physicists do (Kozhevnikov & Hegarty, 2001).

Speed-Accuracy Trade-Offs

Any broadly inferential process must make a trade-off between speed and accuracy (see Heitz (2014) for a review). To illustrate, suppose you were required to judge which of two only very slightly different lines was longer. All other things being equal, making a faster judgement would involve being less accurate, and being more accurate would require making a slower judgement.²

But how can you trade accuracy for speed?

Kozhevnikov and Hegarty suggest that speed can be gained by relying on a simpler model of the physical:

To extrapolate objects’ motion on the basis of [e.g. Newtonian] physical principles, one should have assessed and evaluated the presence and magnitude of such imperceptible forces as friction and air resistance […] This would require a time-consuming analysis that is not always possible. In order to have a survival advantage, the process of extrapolation should be fast and effortless, without much conscious deliberation. Impetus theory allows us to extrapolate objects’ motion quickly and without large demands on attentional resources.’ (Kozhevnikov & Hegarty, 2001, p. 450)

This is one reason why it would be (or is) valuable to have distinct, independent systems. By using different models of a domain (e.g. impetus mechanics vs Newtonian mechanics in the physical domain), different systems can enable radically different, and complementary, trade-offs between speed and accuracy.

Historical Context for the Vertical Motion Example

Moletti (2000, p. 147), who was Galileo’s predecessor in mathematics at Padua, reports an early (1576 or earlier) experiment on the motion of objects launched vertically in a dialogue:

‘PR. […] Aristotle gave rise to doubts by saying that through one and the same medium the speed of things that are moved in natural movement, being of the same nature and shape, is as their powers. That is, if we were to let fall from the top of a tall tower two balls, one of twenty pounds of lead and the other of one pound, also of lead, that the movement of the larger would be twenty times faster than that of the smaller.

‘AN. This seems sufficiently reasonable to me; in fact, if I were asked I would grant it as a principle.

‘PR. You would be mistaken; in fact, both arrive at one and the same time, even if the test were done not once but many times. But what is more, a ball of wood, either larger or smaller than one of lead, let fall from the same height at the same time as the lead ball, would descend and touch the earth or ground at the same moment in time.’

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