Taking Physics Seriously

Tim Maudlin, suggesting in chapter 3 of The Metaphysics Within Physics that gauge theories tell us that current philosophical ontology needs a revamp:

Fiber bundles provide new mathematical structures for representing physical states, and hence a new way to understand physical ontology. For example, modern electromagnetic theory holds that what we call the ‘electromagnetic field’ just is the connection on a fiber bundle. Such an account evidently carries with it quite a lot of ontological commitments: there must be a base space, and internal degrees of freedom at each base point represented by a fiber, and a unified object that corresponds to all of the fibers with a connection. But if one asks whether, in this picture,the electromagnetic field is a substance or an instance of a universal or a trope, or some combination of these, none of the options seems very useful. If the electromagnetic field is a connection on a fiber bundle, then one understands what it is by studying fiber bundles directly, not by trying to translate modern mathematics into archaic philosophical terminology. If an electromagnetic field is a connection on a fiber bundle, then there are more things in heaven and earth than dreamt of in Plato’s or Aristotle’s philosophy. And surely this is to be expected: it would be a miracle if the fundamental ontological structure of the universe fit neatly in the categorical pigeonholes handed down to us from two millennia ago.

The fan of philosophical ontology (universals, tropes, natural sets, or whatever) can no doubt find a way to jerry-rig an object that can do the same mathematical work as a fiber bundle, just as the determined relationist can accommodate the Triangle Inequality by positing it as a separate law of nature (or of ontology?). The question is what the point of such an exercise could possibly be. Fiber bundles have their own interesting structure, a structure that does not correspond to the traditional philosophical vocabulary. And the structure of the physical world might be that of a fiber bundle. So instead of a rearguard operation in defense of philosophical tradition, philosophers would do best to try to understand the structure on their own terms. If they do not translate well into the categories in which philosophical debates have taken place, so much the worse for the philosophical debates.

Richard Healey points out in his review of the book that taking a gauge field to be a connection on a fiber bundle is a mistake. But that doesn’t detract from Maudlin’s main point, that the structure of the world suggested by modern physics is rather too complex for the philosophical terminology that is in vogue. Are there good arguments for the person who wants to make it a requirement that physical theories be assimilated into the simple concepts used in our current metaphysical terminology?

Philosophy on YouTube

Someone has uploaded videos of Bryan Magee’s discussions with assorted famous philosophers. I just watched Searle on Wittgenstein; it would make a pretty good introduction to Wittgenstein’s philosophy, even if he (inevitably) says a few controversial things.

“Strong Impingement” in Musical Experience

More Stream of Consciousness blogging. In chapter 8, Dainton defines Strong Impingement as the thesis that

Phenomenal wholes have certain parts that possess intrinsic phenomenal features that reflect the character of that whole, and parts with the same character could not possibly occur except in a whole of the same or similar type.

He thinks that examples of strong impingement are at most extremely rare in our experiences. However, I think he really steps into it when he discusses, as one of the first few examples in his evaluation of strong impingement, the phenomenology of music. He writes:

The individual tones that make up a melody are heard as parts of a whole that evokes various responses in us. We might recognize that the notes form a melody without recognizing the melody as one we have heard previously. Or the melody might be familiar: as we listen we can anticipate the notes to come. In either case, the melody might produce affective and aesthetic responses, for example we find it beautiful but sad, or pleasurable but hackneyed. But as in the visual case, these responses involve our broader state of mind — they are not confined to the phenomenal characteristics of our perceptual experience — and they are produced by the melody as a whole, as it unfolds, rather than any particular note. Consequently, it remains unclear to what extent, if any, the intrinsic phenomenal characteristics of the constituent notes would be different if they were heard in isolation.

So Dainton doesn’t want to deny that something about our experience as a whole is altered by holistic aspects of our experience. But he doesn’t think what is altered can be said to be any of the “intrinsic phenomenal characteristics” of the parts of the experience.

This line doesn’t work too badly for visual experience. Dainton uses the example of an isolated eye versus the same eye framed by a cow’s head. Obviously the presence of the whole “cow’s head” experience alters our perception of the eye in some way — it is a now a cow’s eye, not just a disembodied eye, and we’re likely to unconsciously attribute cow-associated characteristics to it (passive, bovine, who knows). But Dainton would say that such attributions do not form part of the “intrinsic phenomenal characteristics” of the eye-experience. Instead, they are simply part of our overall mental state as we contemplate the eye.

I have my reservations about this claim, but these reservations turn into vehement objections when it comes to musical experiences. Consider any harmonic progression, let’s say a cadential progression, I-II-V-I. I would say that the strongest phenomenal characteristic of the V chord is not the phenomenal characteristics of the notes themselves (their pitch, timbre, loudness, etc.), but the “penultimate-ness” of the dominant harmony. A listener with perfect pitch may also be aware of the pitch-characteristics of the chord, and most listeners are going to be at least vaguely aware of the timbre, but the penultimate-ness would be stronger than all these other characteristics — and I’m not just saying this from a music theory perspective, but from a phenomenological perspective — the tension and drive towards the tonic that a listener would experience is far stronger than any of the other phenomenological characteristics of the chord. And the dominant-chord-like quality of V is of course due to the experience of the whole chord progression — alter some of the other chords, and we could easily change that quality to something quite different, say a tonic-like-quality (stability), or a dissonance (alien-ness).

Now this still doesn’t strictly rule out Dainton’s probable claim that the emotions we have upon the entrance of the dominant are simply due to our “mental background” and should not be attributed to the phenomenological characteristics of the chord itself. An extremely strong phenomenal quality could still be attributed to one’s “overall mental state”, rather than the experiental part itself. But let’s look again at Dainton’s reason for concluding that the emotional aspect of melodies aren’t “intrinsic phenomenal characteristics” of the components of melodies: “these responses involve our broader state of mind — they are not confined to the phenomenal characteristics of our perceptual experience — and they are produced by the melody as a whole, as it unfolds, rather than any particular note.” (Emphasis mine).

I think it is uncontroversial that the “melody as a whole” is a necessary context for us to experience the particular note as it is experienced — the note itself cannot do that. But the question is not the causal one of what is responsible for the particular note sounding (say) tense. It is whether we should attribute the tenseness we hear in the note to our “broader state of mind”, or to the note. To me, it is intuitively obvious that it should be the note. This is not something I feel to be in my overall mental state. It is not an unlocalized, pervasive presence. When the bittersweet Neapolitan 6th chord sounds, I want to point to it and say, that sounds bittersweet. It’s not that I feel bittersweet (I might, though). Instead, I am perceiving a phenomenal object in my experience to be bittersweet. And the phenomenal object is that chord. Or that cheerful motif. Or that dissonant, ominous trill in the bass. This localization of the experienced phenomenal quality cannot be explained away by any theory that says it is merely part of my overall mental state.

So I would conclude that Strong Impingement holds for musical experiences, to a significantly greater, or at least more obvious, extent than for visual experiences. And part of the reason why it seems to be important in musical experiences is because of the way meaning in music is created. Meaning in music is far more contextualized than meaning in visual experience (let us, for the moment, discount visual experiences containing linguistic images). For the class of humans who know what an eye is, we can imagine them having a “core” type of phenomenal experience (but not identical experiences) whenever they encounter eyes — it’s easy to recognize an eye as an eye outside of its usual position in the faces of animals. But it’s outright impossible to recognize a dominant chord as a dominant chord outside of its surrounding harmonic context. The intrinsic phenomenal characteristics of a chord are necessarily a product of its context, whereas it’s quite likely that that may not be necessarily (but could be contingently) so of a visual phenomenal object.

One is then almost inevitably led to the question of whether Strong Impingement holds for linguistic experience (not counting music as a language for now). Dainton thinks not, for largely the same reasons that he thinks it doesn’t hold for visual experience. And I think he’s right to judge linguistic experience similarly to visual experience in this respect, because most parts of our linguistic experience have strong phenomenal characteristics (by association) by themselves. Again, contrast this with music, where two identical chords, which by themselves are phenomenologically the same, can have vastly different phenomenological effects when placed in different contexts. This is true for some words in natural languages, e.g. “bank” in the English language. Dainton does consider “bank” in his discussion of linguistic experiences:

But even in this sort of case, the influence of context is limited: the sound or inscription ‘bank’ means either ‘place or institution where money is deposited’ or ‘the side of a river’; context merely makes it clear which of these meanings is intended.

I have to confess, though, that I don’t see how this resolves the issue. It’s true that the word “bank” means either financial institution or side of river, but we almost never experience it with the disjuncted meaning “either financial institution or side of river”. In all but a few cases of contextual ambiguity, we experience the sound or sight of the word “bank” as indicating exclusively one or the other of its two possible meanings, not as a two-valued ambiguity.

Cases like the word “bank”, though, are relatively less common in natural languages (in comparison to the “language” of music). I’d say linguistic experience is between musical and visual experience on the spectrum of experiences to which Strong Impingement applies.

Parochial Piano-playing Thoughts

I’ve been trying to master the second movement of Schubert’s last piano sonata (in B flat, D960) for a few months now. It’s one of my favourite slow movements ever, because it breathes (not just describes) loneliness and desolation. A feature common to the first two movements of D960 is the intrusion of an alien-sounding motif in the bass. In the first movement, the motif provides an ominous contrast to the otherwise relatively upbeat lyricism of the main subject. In the second movement, which has an ABA’ structure, the motif is completely absent from the A section, but is pervasive in the A’ section. Here are the first bars of both sections:

Its ‘intrusion’ is one of the main reasons why the A’ section sounds significantly more ‘hopeless’ than the A section. The word ‘intrusion’ might sound inappropriate at first, but if you listen to both movements, the bass motif sounds as though it doesn’t belong. Before I started playing the piece myself, I’d assumed that it was simply because it wasn’t expected (not having appeared previously, at least not in the same prominent context), and because it was in an unusually low register (in contrast to the left hand’s rising octaves in the second movement). But I find that I’m having difficulties making it sound alien when I play it. Perhaps it’s because I’m playing it, so I can no longer fool myself into thinking of it as an intruder, especially when my left hand is dutifully keeping time with my right hand as it plays it. It sounds a little more alien when I play it more staccato and more stiffly, but still nowhere as alien as I expected and want it to sound.

On a more minor note, every performance I’ve heard of this piece significantly ups the tempo for the B section — understandably, since if they played the B section at the same tempo as the A section, it would sound unsuitably ponderous. But my score, at least, does not indicate any tempo change for the B section, and the acceleration when they transition from A to B always sounded a bit weird to me. I’ve always wondered if perhaps the outer sections are simply being played too slowly, or at least more slowly than Schubert had intended.

As lovely as D960 is, on days when I feel like doing something rather more vigorous and angry, it doesn’t suffice. That’s when I return to a section of Schubert’s Fantasy in F minor, a piece I’d essentially stopped practicing after my duet partner left. There’s something incredibly satisfying about having your left hand rage with the strident, accented melody while your right hand unleashes flurries of triplets:

Against a relationist conception of phenomenal space

Chapter 3 of Barry Dainton’s Stream of Consciousness explores phenomenal space, which is the space[s] in which our experiences seem to take place. Dainton tries to extend the absolutist/substantivalist-relationist debate about physical space to phenomenal space. Here’s what Dainton means by a absolute/substantival phenomenal space:

for a phenomenal space to merit the label ‘substantival’, the space in question should possess some phenomenal reality — the space in question must possess some intrinsic phenomenal characteristics of its own.

For example, a visual field that has a luminous white quality throughout except where objects appear would be a substantival phenomenal space. Our auditory field, Dainton thinks, is an example of a relationist phenomenal space — it is a ‘phenomenal void’ everywhere except where sounds exist.

I was highly sympathetic to the relationist position on phenomenal space at first, for the same reasons that I’m sympathetic to relationist positions on physical space: if we discard all physical apparatus that enable us to sense direction (the fluid in our ears, our knowledge of which direction things tend to fall in, our knowledge of left versus right, etc.), then how could we possibly know if a flag (say) in our visual field is upside-down?

But upon further reflection, I think phenomenal motion makes it very difficult for us to accept a relationist conception of phenomenal space, at least for visual phenomenal space. If phenomenal space is defined entirely by the relations between objects occurring in it, then a phenomenal space in which (in absolutist-speak) there is a sole rigid object moving with a constant velocity would be the same phenomenal space as that in which there is a sole stationary rigid object. But it seems highly counter-intuitive to think that we cannot, in our experience, see the difference between utter blackness with a sole moving object and utter blackness with a sole stationary object. Could it be that my intuition that we can see such a difference is due entirely to my having fluid in my ears and other kinaesthetic aids that make motion noticeable to me in a way that isn’t due to visual phenomenal space per se? But what has the fluid in my ears got to do with whether other objects, independent of me, are moving? The fluid in my ears, and my other kinaesthetic aids, would seem to tell me only about how my body is oriented, not about how other bodies are moving.

It seems likely, too, that part of the reason why we can see that an object in a black void is moving is because our visual field is finite, so we can actually see objects move to and from its ‘edges’. Granted, they aren’t very well-defined edges, but they are there, and they seem to form a background frame against which we can judge if something is moving. So there already seems to be a delimited space ‘out there’ in which objects are situated. When objects are added to it, they are in some situation nearer or further from the edges — they don’t define those edges.

A Real-Life Carrot?

Perhaps it’s only because I haven’t read any Terry Pratchett for some months that it took so long to strike me, but I’ve just noticed the striking resemblances between Carrot Ironfoundersson and Barack Obama. Both are charismatic and give the impression that they ‘understand’ people, even their supposed enemies, better than these people understand themselves (even David Brooks, hardly an Obama supporter, acknowledges this). Both enjoyed quick rises through their respective professions. Both work hard to ‘unite’ people amidst racial (in the case of Carrot, inter-species) tensions. Both attempt to respect their opponents’ positions where other partisans would make straightforward demonizations (again, Brooks testifies to this). Both give off an appearance of such sincere optimism that most people suspect that it’s just a facade. The Wikipedia article describes Carrot as often taking time “to see all sides of a story before getting involved wholeheartedly”. Well, Cass Sunstein is simply one of many who have described Obama as doing the same.

There are a few other parallels that have less to do with personality, but perhaps are all the more suggestive for that. The most obvious is that of possible national leadership: Carrot is the unacknowledged heir to the throne of Ankh-Morpork. A less obvious point, but one that is important to the identities of both Obama and Carrot, is the complex and somewhat mixed racial (in Carrot’s case, multi-species) roots of both. Carrot is biologically human but was brought up as an orphan by dwarfs. Culturally, he identifies with dwarfs, and is able to defuse tensions involving Ankh-Morpork dwarfs for this reason (the ‘uniter’ theme creeps back again).

Part of the awe that perhaps even people ‘outside’ the Obama cult have for Obama is that they cannot believe that Obama could really be what he professes to be and that a simple naive message of the kind he is spreading could actually work. This “Is this guy for real?” sentiment is also often directed at Carrot as he goes about his business in Ankh-Morpork. Well, Carrot’s way did work. If Obama’s does as well, then chalk one up to the sophistication of Discworld characters and settings — and to Pterry‘s genius.

Some JEB commentaries on group versus kin selection

Ever since I read The Selfish Gene, I’ve been wondering what exactly group selection could bring to the table of theoretical population genetics that kin selection could not. Every case of group selection observed in nature that I read about turned out to involve groups of individuals that have a higher-than-average relatedness within their group (compared to the rest of the population). So why are the group selectionists still going strong?

A recent exchange of commentaries in the Journal of Evolutionary Biology, all free access, seemed like they could go some way to answering my question. They started out with West, Griffin and Gardner putting forth a new classification system for social behaviours, particularly those that are broadly regarded as ‘altruistic’. They devote one section of their paper towards ironing out the ‘semantic confusion’ created by the group selection literature. First, they distinguish the ‘old’ and ‘new’ group selection theories. The old one, associated with Wynne-Edwards, treated the group as the only level of selection — behaviour that was ‘for the good of the group’ is selected for, and groups were selected at the expense of other groups — selection was interdemic. The ‘new’ group selection, in contrast, involved intrademic selection: it considered small groups of individuals that had mutual interactions, and showed that cooperative behaviour within those small groups could be favoured. West et al put it this way:

the new group selection approach looks at the evolution of individual characters in a group structured population, whereas the old group selection approach looks at the evolution of group characters

West et al point out that the ‘old’ group selection works under only extremely restrictive conditions. The ‘new’ group selection, on the other hand, can be interpreted as kin selection — group-selectionist and kin-selectionist descriptions of it are mathematically identical. The implicit suggestion is that there is no point in keeping the group-selectionist description: why not leave everything to kin selection and individual selection?

Unsurprisingly, prominent group selectionist David Sloan Wilson objects to the above account of group selection. Strangely, his reply involves a lengthy recap of the history of group selectionist ideas. Less strangely, he uses the catch-phrase ‘pluralism’ all the time.

Wilson doesn’t try to deny that every plausible case of group selection discovered so far can be reinterpreted as kin selection. He maintains, however, that group selection has brought new insights to population genetics, citing the examples of population viscosity and human cooperation. In particular, he claims that group selectionist perspectives were important in originating modern lines of research on those two issues.

In their reply to Wilson’s commentary, West et al concede that group selection made a pioneering key contribution to models of population viscosity. Their comeback, however, is that later approaches using kin selection models were able to provide analytical proofs that Wilson’s group selection approach could not. On the issue of whether the effect of local competition can be overcome by dispersing in small groups (‘buds’), they point out that kin selection models have managed to provide a solution with just “a few lines of algebra”, whereas group selectionists had long lamented that the problem was too mathematically complicated.

In response to Wilson’s charge that group selectionist perspectives provided new insights into human cooperation, West et al have a similar reply: briefly, that group selectionist approaches have not been able to derive analytical solutions where kin selectionist approaches were able to. They further charge that on this issue, “the group selection approach has failed to clarify the underlying selective forces, and has led to confusion”. They believe that kin selection better isolates these ‘underlying selective forces’. For example, kin selection has demonstrated that “punishment or strong reciprocity are not alternative evolutionary explanations for cooperation, as had been implied, but merely specific mechanisms for providing direct or indirect fitness benefits to cooperation” — these ‘fitness benefits’, rather than the punishment or strong reciprocity in themselves, are the ‘underlying forces’ of selection.

As I’d hinted at earlier, I was uncomfortable with Wilson’s heavy citation of history in his defence of group selection. For even if group selection has historically been a pioneering approach to many issues in evolutionary biology, it doesn’t follow that it’s still worth keeping around as an alternative approach. To show the latter, you really have to bring up examples of issues in which group selection has produced results that kin selection has not been able to. And it seems that West et al have responded admirably to Wilson’s examples by countering with kin selectionist approaches to them that have not just matched but bettered group selectionist approaches.

It probably reflects my physicist-bias that I also find West et al’s argument that group selection should be dropped for reasons of theoretical unification persuasive. It seems that some element of above-average relatedness between cooperators is always present in successful group selection scenarios. The natural move is then to isolate this element as ‘underlying’ group selection, which is exactly what kin selection does. And even on an instrumental level, kin selection seems to beat group selection. Sure, there was a time when group selection had its nose ahead of kin selection on certain issues. But many abandoned scientific theories also at some point or another were the ‘leading’ approaches in their field. We should embrace pluralism only if it’s going to be useful to us now.

WEST, S.A., GRIFFIN, A.S., GARDNER, A. (2007). Social semantics: altruism, cooperation, mutualism, strong reciprocity and group selection. Journal of Evolutionary Biology, 20(2), 415-432. DOI: 10.1111/j.1420-9101.2006.01258.x

WILSON, D.S. (2007). Social semantics: toward a genuine pluralism in the study of social behaviour. Journal of Evolutionary Biology DOI: 10.1111/j.1420-9101.2007.01396.x

WEST, S.A., GRIFFIN, A.S., GARDNER, A. (2007). Social semantics: how useful has group selection been?. Journal of Evolutionary Biology DOI: 10.1111/j.1420-9101.2007.01458.x

PPC and the Cutaneous Rabbit

There are several parts in Dainton’s Stream of Consciousness that leave me cold because they endorse such a simplistic view of conscious experience that I don’t see much hope for reconciling them with certain counter-intuitive findings from cognitive psychology. One example of an obvious over-simplification that raised my hackles is Izchak Miller’s Principle of Presentational Concurrence (PPC). The relevant Miller definition, as quoted in Dainton, is that PPC proposes that

the duration of a content being presented is concurrent with the duration of the act of presenting it… the time interval occupied by a content which is before the mind is the very same time interval which is occupied by the act of presenting that very content.

Dainton thinks PPC is plausible because:

When I see the red flash being followed by the green flash, or when I hear a sequence of notes C-D-E, my experiencing of the succession does seem to run concurrently with the phenomenal contents which jointly constitute the succession; I am aware of the red flash before I am aware of the green flash. Or so it seems natural to say. To this extent, it is counterintuitive to suppose my awareness of the succession occurs an instant after the succession has occurred (or at the very last instant of the succession).

My contention is that it is difficult to reconcile PPC with the ‘cutaneous rabbit‘ experiments. The original experiment by Geldard and Sherrick found that when five brief mechanical pulses are transmitted to the same point on the wrist, then subsequently another five to a point further up the arm, and the last five to a point near the elbow, the subject does not feel five taps each at three widely separated points on his arm. Instead, he perceives “a smooth progression of jumps up the arm, as if a tiny rabbit were hopping from wrist to elbow”.

In the cutaneous rabbit illusion, it seems that the content must be presented to one’s awareness only after the succession of phenomenal contents has occurred. It is only after the brain has taken in the five taps at a spot further up the arm, that it then ‘goes back’ to ‘fill in the blanks’ of the taps that it thinks must have occurred in between. The brain, surely, cannot concurrently present the phenomenal content of taps occurring in the sequence of positions x, x+1, x+2, x+3, x+4 while the tapping is occurring only at the point x. What the cutaneous rabbit illusion suggests is happening is that the brain waits for the whole sequence of taps to finish before presenting to our awareness its interpretation of what happens. Yet if we terminate the experiment after the first five taps, it is clear that the phenomenal contents are taps at the point x, not taps that proceed up the arm.

The only way out for the supporter of PPC here is to deny that the phenomenal contents are the taps administered by the device. He could say instead that the phenomenal contents simply are what one is aware of — the sequence of closely spaced taps hopping up the arm. But that does not mesh well with the fact that if we consider only the first five taps at the wrist, there’s no way that these can be concurrently apprehended as taps moving up the wrist, unless the brain has a premonition that the next five taps will be occurring at a point much further up the arm. At the time of tapping, the first five taps must simply be experienced as taps on the wrist. Concurrent awareness just isn’t consistent with the obvious post-editing of experiences the brain makes before presenting the whole experience to our awareness.

On Dainton on Broad

More slogging through Dainton’s Stream of Consciousness. As the book progresses I become less sympathetic. He relies on conclusions from earlier chapters to advance his arguments in the later chapters, and my lack of sympathy for the various early conclusions compounds to produce an overall significant deficit of sympathy for his later conclusions.

Here’s a brief objection to one of his criticisms of Broad’s later account of our experience of time. On page 148, Dainton writes:

So far as [Broad's earlier account] is concerned, it is literally true that we are directly aware of sensible motion and rest, for immediate experience extends over time. According to [Broad's later account], no change or duration can be experienced as fully present, since maximum presentedness is possessed only momentarily. This is phenomenologically suspect.

Here it seems that Dainton is neglecting altogether an important distinction: that between the length of time our experiences appear to us to occupy and the actual length of time occupied by the experience. We all know that particularly boring stretches of time can appear to be longer than the actual time they take up. When Broad says that ‘maximum presentedness’ can be possessed only momentarily, it seems that he is referring to the actual length of time occupied by the experience of maximum presentedness. However, the phenomena that are presented to us with maximum presentedness may themselves appear to us as non-momentary happenings. That is, a experience that is phenomenologically five seconds long may still be presented to us instantaneously in ‘real time’, with maximum presentedness, so that we do experience duration ‘momentarily’.

What’s more curious about Dainton’s neglect of a pretty obvious distinction in his criticism of Broad is that he does make that distinction in the next chapter, when outlining John Foster’s ‘overlap model’ of temporal experience:

Once we recognize a temporality that is intrinsic to phenomenal contents, we can — at least in principle — distinguish this time from the time at which these contents are sensed, i.e. presentational time. In brief, phenomenal time is the time in experience, presentational time is the time of experience.

So I don’t know if I’m misreading his criticism of Broad, if perhaps he isn’t claiming there that a momentary experience in presentational time must be one of a moment in phenomenal time.

Addendum: It seems that Dainton recognizes the distinction between phenomenal time and presentational time in principle, but still thinks (inexplicably) that time in experience has to be of the exact same length as the time taken up by the process of experience. On p181, he criticizes the tendency to think of temporal events as though they are spread out in space and can be apprehended, as a spread, momentarily. In the midst of his criticism he makes the following assertion: “Experiencing a temporally extended process requires time — just as much time as the process itself takes up.” However, he doesn’t make any argument for this assertion. He is probably right that there is little reason to trust analogies between temporal awareness and spatial awareness, but that in itself isn’t enough to support the assertion just quoted.