Monday, 2 October 2017

Toomas Karmo: Part O: Philosophy of Perception, Action, and "Subjectivity"

Quality assessment:

On the 5-point scale current in Estonia, and surely in nearby nations, and familiar to observers of the academic arrangements of the late, unlamented, Union of Soviet Socialist Republics (applying the easy and lax standards Kmo deploys in his grubby imaginary "Aleksandr Stepanovitsh Popovi nimeline sangarliku raadio instituut" (the "Alexandr Stepanovitch Popov Institute of Heroic Radio") and his  grubby imaginary "Nikolai Ivanovitsh Lobatshevski nimeline sotsalitsliku matemaatika instituut" (the "Nicolai Ivanovich Lobachevsky Institute of Socialist Mathematics") - where, on the lax and easy grading philosophy of the twin Institutes, 1/5 is "epic fail", 2/5 is "failure not so disastrous as to be epic", 3/5 is "mediocre pass", 4/5 is "good", and 5/5 is "excellent"): 4/5. Justification: There was enough time to write out the  necessary points to reasonable length.

Revision history:

All times in these blog "revision histories" are stated in UTC (Universal Coordinated Time/ Temps Universel Coordoné,  a precisification of the old GMT, or "Greenwich Mean Time"), in the ISO-prescribed YYYYMMDDThhmmZ timestamping format. UTC currently leads Toronto civil time by 4 hours and currently lags Tallinn civil time by 3 hours.

  • 20171003T1610Z/version 3.0.0: Kmo finished converting his point-form outline into coherent full-sentences prose. He reserved the right to make tiny, nonsubstantive, purely cosmetic, tweaks over the coming 48 hours, as here-undocumented versions 3.0.1, 3.0.2, 3.0.3, ..., .   
  • 20171003T0311Z/version 2.0.0: Kmo uploaded a correctly detailed, correctly polished, point-form outline. He immediately made a bare start on converting this into coherent full-sentences prose. He planned then to get some sleep, and to finish the job of conversion the following morning, by 20171003T1600Z or so. 
  • 20171003T0008Z/version 1.0.0: Kmo, running a bit behind schedule, was able to upload an only partially  polished outline, with its latter part not just unpolished but even lacking in details. He hoped to convert this work into an everywhere correctly detailed, and everywhere correctly polished, outline at some point in the next two hours, and  immediately after that working session to make at least a start on the task of converting the outline into coherent full-sentences prose.]

[CAUTION: A bug in the blogger server-side software has in some past months shown a propensity to insert inappropriate whitespace at some points in some of my posted essays. If a screen seems to end in empty space, keep scrolling down. The end of the posting is not reached until the usual blogger "Posted by Toomas (Tom) Karmo at" appears. - The blogger software has also shown a propensity, at any rate when coupled with my erstwhile, out-of-date, Web-authoring uploading browser, to generate HTML that gets formatted in different ways on different downloading browsers. Some downloading browsers have sometimes perhaps not correctly read in the entirety of the "Cascading Style Sheets" (CSS) which on all ordinary Web servers control the browser placement of margins, sidebars, and the like. If you suspect CSS problems in your particular browser, be patient: it is probable that while some content has been shoved into some odd place (for instance, down to the bottom of your browser, where it ought to appear in the right-hand margin), all the server content has been pushed down into your browser in some place or other. - Finally, there may be blogger vagaries, outside my control, in font sizing or interlinear spacing or right-margin justification. - Anyone inclined to help with trouble-shooting, or to offer other kinds of technical advice, is welcome to write me via]

Last week, in "Part N", I considered the limits of my agency, remarking that two different things are in principle conceivable. On the one hand, it could in principle be that I draw the bounds too widely, by regarding myself, in the manner of "Finkelstein's" wind-blown leaf, as executing various performances I do not in fact as an agent execute. On the other hand, it could in principle be that I draw the bounds too narrowly. On this scenario, I absentmindedly do various things which I do not, in my current state of absentmindedness, presently notice myself doing - as when, in a sort of logical inversion of "Finkelstein's" windblown leaves, I become (engrossed as I unfortunately am in the Sherlock Holmes railway chase scene) responsible for the to-and-fro movement of some litter on the cinema floor.

Indeed I may as well add tonight that one might do these various things absentmindedly without ever, at any point at all, waking up to the fact. For instance, one could first be absentmindedly moving litter back and forth in the cinema, and immediately thereafter be swooning in the excitement of the depicted railway drama - even, for all that logic can say, entering a permanent and unending swoon - without noticing what it is one has been doing with one's feet.


Although some readers will rightly wince at pedantic repetition, it may perhaps help other readers if I say it once again (now amplifying just a little): I am not, in my occasionally outlandish Gedankenexperimente, denying that the bounds of my agency are what common sense, or again common sense as moderately refined by neorophysiological science, assert them to be. In particular, I am not denying the truth of the commonsense beliefs that I exercise (for instance) agency over the Toomas Karmo hands, and that I myself lack agency over (for instance) the Donald Trump hands. Again, I am not denying the truth of the belief, due to common sense as moderately refined by physiological science, that I exercise (for instance) agency over the Toomas Karmo biceps and the Toomas Karmo motor cortex, while lacking agency over (for instance) the Donald Trump biceps and the Donald Trump motor cortex.

I additionally concede, even while describing outlandish Gedankenexperimente, that it is in some sense rational to believe that I exercise agency over the Toomas hands, biceps, and motor neurons, and lack agency in respect of the Donald hands, biceps, and motor neurons. (Admittedly, I do not know how to articulate the sense in which such beliefs are "rational", beyond making banal and superficial points about the "rationality of taking seriously the anatomy textbooks, given what other things I on good authority believe about publishing houses and anatomy professors".)

Indeed (this counts as a bit of amplification, going above and beyond what I guess I have written in previous weeks), I affirm that I know myself to exercise agency over the Toomas hands, biceps, and neurons, and know myself to lack agency in respect of the Donald hands, biceps, and neurons. (To be sure, I seem unable to answer, except in the barest and most banal outline, the persistent philosophical skeptic who asks, "How is it you know?")

Some of my readers, not accustomed to the on-occasion-careful ways of modern Departments of Philosophy, might possibly at this point ask, "Why harp over both rational true belief and knowledge? Is not knowing that p the same as having a rational true belief that p?" But the harping does have some (modest) point. Here is how they would put it in a modest, downmarket, modern Department of Philosophy, at my imaginary downmarket Tallahassee Swampwater Junior Training College. (This is a Department where they retail clichés for decades familiar from the philosophical journals, and do on occasion manage to be commendably careful; Swampwater, as schools go, is modern in at any rate a 1970s, polyester-and-vinyl-and-styrofoam, video-recorder-and-colour-television, sense.) You are standing on the familiar railway platform, looking up, for the umpteenth time in your life, at the familiar 12-hour clock. It is rational for you to trust that clock, for never in your long past has it let you down. So, trusting it as you habitually do, and noting that its little hand is at 8 and its big hand just to the right of the top of that 12-hour dial, you form the belief that the time at this point in your familiar morning commute is just after 8.00 - say, 8.01 or 8.02. Unknown to you, however, the clock has, for the first time in twenty long years, malfunctioned. It stopped last night, just after 20.00. It is now showing the correct time, in the usual 12-hour notation, by sheer accident. So whereas you think you know the correct time, you do not in reality know it. Rational true belief is one thing, and knowledge is a different thing. Although the two states of mind often go together in the real world, they do need to be analyzed separately. 


It is now time to examine more closely last week's notion of absentmindedness.

Absentmindedness in action is common. Many people do not know about motor neurons - or more accurately, having seen something on the BBC or PBS a while back, they are now too busy on legitimate pursuits to reflect further. Such people can be said to be "absentmindedly" firing their neurons, even while being duly mindful of the movements they are making with their hands. Such busily practical people are typically, again, mindful of such necessary things as motorcar steering wheels and motorcar turn-indicator lamps, or again of computer keyboards, and again of pencils and papers - causally linked as vehicles, keyboards, pencils, papers, and the like are in our so-mechanized contemporary society to guiding human hands.

The sense in which practical-minded people are mindful in their actions is brought out by the late Prof. G.E.M. Anscombe (1919-2001) in her 1957 book Intention. (Some readers will be helped by my  having herewith flouted, yet again, my Igominy and Humiligation Precept from "Part B", back on 2017-05-22 or 2017-05-23.) Humans often know, says Prof. Anscombe, in a direct kind of way what they are doing. They know this, she writes, not on the basis of looking at themselves. Even without looking, for instance, you, the Gentle Reader, know that you are raising an arm. (The performance is unlike levitating your arm by uttering "Om, om, om," while glancing anxiously at your fleshy upper-torso appendage to ascertain whether it is currently down or up. The performance just imagined would be appropriate if you had, rather, mental powers analogous to those claimed by telekinesis pioneer Yuri Geller, and had suffered an injury to the nerves linking biceps with motor cortex.) In the spirit of Prof. Anscombe, I would add that these days one even knows, without bothering to look at screen or keyboard,that one is inputting such a sequence of letters as t, h, and e. (It is common enough to type, for instance, a password, without looking at the keys, and in the knowledge that the password is for cybersecurity reasons not being echoed to the screen.)

The raising of an arm, or again the blind touch-typing on a busy day of some duly simple word with duly habituated fingers, is in normal human circumstances a clear example of mindful doing - even though absentminded arm-raising, and again absentminded password-typing, might on occasion also occur.

In a parallel way, the seeing of sunlit grass is in normal human circumstances a clear example of something undergone mindfully, or at least undergone not quite mindlessly. It is surely often like this, that our attention is focused on some picnic hamper, or on some recreational paperback, or on some bird or squirrel, and that we are nevertheless moderately mindful of seeing the sunlit grass that so-to-speak frames the item of central current perceptual interest.

We are perhaps more absentminded about our greening - perhaps not bothering to form the concept of greening (as distinct from the so-familiar concept of seeing grass) until it is drawn to our notice in some philosophical essay such as this one.

We are likewise more absentminded about our seeing retinal patches-of-green-light, and about our seeing neuronal events.

I have already to a considerable extent said - but this week I say it again - that it seems to me a merely contingent matter, a so-to-speak merely medical matter, what things we perceive and do with mindfulness and what things we perceive and do absentmindedly.

Is it at least a conceptual, logical, necessity that somewhere in the long perceptual chain (T-ing in U-ing in V-ing in...), and somewhere in the long, logically rather similar, agency chain, there is something mindful? With some diffidence, I say "no" even to this. (Readers some day feeling themselves able to argue a contrary, convincing, "yes" might want to e-mail me, via the usual address As far as I can at the moment see, it is possible for a perceiving and acting person to be thoroughly and radically absentminded - as, perhaps, in the case of a cloistered religious, or again of a university mathematician, in a contemplative ecstasy, who sees the chapel flagstones or the Fields Institute walls, and who grips the pectoral cross or the library-chair armrest, while becoming so rapt in her or his thoughts (regarding, as it might be, the Incarnation, or again Gauss's theorem relating volume-pervading Divergence to integral-of-flux-through-bounding-surface) as to forsake mindfulness-in-perception and mindfulness-in-agency. Such a state of contemplative ecstasy would be unlike an outright swoon, in which perception and agency in a banal way cease. No - seeing and acting continue, albeit in a temporary state of religious or mathematical self-forgetting.


A full "Geography of the Mind" requires a little more than just attention to these principal twin themes of perception and action. We not only perceive and act: there are also (I) occasions on which we imagine (both in a perceptual and in a practical (Anscombe-agentual) sense), and (II) occasions on which we think - in a special, tricky, deep-and-narrow sense of "think" on which thinking is other than perceiving (be it ever so laudably mindful), other than acting (be it ever so laudably mindful), and even other than imagining. I may as well, in the interest of Geography-of-the-Mind completeness, (I) make a few remarks here on imagining, and then (II) set a modest piece of homework on thinking.

On the perceptual side, each of the following is a possible perceptual report from you, the Gentle Reader:

  • "I imagine seeing breeze-swept, waving, grass."
  • "I imagine seeing a hand's rising." 
  • "I imagine seeing a tiny patch of green light [or pinkish light] sitting near the top of [or descending toward the middle of] a retinal surface, as my Homo sapiens converging ocular lens projects a tiny (inverted) image from lawn and rising hand." (In the sense I have in mind here, this bit of seeing is imagined not in imagining a visit to an anatomy lab where an eyeball is opened to inspection from outside, but merely in imagining, e.g., the seeing of a lawn behind a rising hand. Admittedly, one can absent-mindedly, being fixated on grass and hand, fail to note that this rather recondite retinal thing is also, indeed is eo ipso,  being imagined, as so-to-speak "an inspection from inside".) 
  • "I imagine seeing a visual-cortex neuronal event." (This, again, is imagined, in the sense here being called to mind, in imagining, e.g., the seeing of grass. To be sure, one can once again absentmindedly, when fixated on grass, fail to note that this rather recondite cranial thing is eo ipso being imagined.) 
  • "I imagine greening," (It is unfortunate that English - unlike, I kinda-sorta think, Biblical Hebrew - lacks a distinction between active-sense single-word present participles and single-word passive-sense present participles. Is this, or at least something close to this, marked in Biblical Hebrew by Qal-and-Piel-and-especially-Hiphil on the active side, and Niphal-and-Pual-and-especially-Hophal on the passive side? At the moment, my Hebrew is in the position of a full-year seminary-or-Classics course that is just some modest distance into the second semester. The nuances of the just-mentioned Hebrew stems come, alas, later on the course, as it were when the winter thaw is at hand, with March Break or University of Toronto "Reading Week" fast impending. Doing what I can with English, I remark that I have here in mind a passive (a Hophal?) sense, on which "I green/thou greenest/she-or-he greeneth" parallels the passive "I thirst/thou thirstest/she-or-he thirsteth." - Since I am likely sooner or later to have a few Estonian-language readers, I do remark that the requisite distinction is with some reasonable measure of success marked in our native tongue, with a pair of rather mild neologisms - Ma kujutan ette ome rohetumist for the intended passive sense, and Ma kujutan ette oma rohetamist for the contrasting, in-this-list-not-intended, active sense.
On the side of agency (the "active" side), the following are possible autobiographical reports from you, the Gentle Reader:

  • "I imagine raising a hand."
  • "I imagine firing motor-cortex neurons." 
Further, I have in  previous installment had us imagine a false thing on the active side (as one could on the passive side imagine, falsely, seeing a unicorn):

  • "I imagine waving the grass."
Again, one can contemplate the counterfactual possibility of possessing what Homo sapiens lacks, namely the Voluntary Visual Cortex (cf "Part M", from 2017-09-04 or 2017-09-05). Although unicorns are never seen, the seeing of a unicorn is readily imagined. Analogously, although no human is in a position to say, truthfully, "I am greening" (in the active, Qal-Piel-Hiphil, Mina rohetan as opposed to Mina rohetun, sense), humans possessing a powerful Practical Imagination can truthfully say, "I am imagining a practical (agentual) falsehood, namely doing-some-greening" - Ma kujutan ette oma rohetamist (as distinct from the perceptual-imagination Ma kujutan ette oma rohetumist).  Here is a further example of imagining a Practical Falsehood, once again akin to that Perceptual Falsehood which is imagining seeing a unicorn: "I imagine making myself, contrary to Matthew 6:27, swiftly expand, pushing the crown of my head a full cubit closer to the clouds."

In short, just as there is a "Practical Knowledge", through which one knows even without looking that one is, e.g., raising a hand, or is typing the letters t, h, and e as part of a screen-concealed password, so too there is a "Practical Imagination", separate from the more often discussed "Perceptual Imagination". To imagine yourself raising a hand, you need not imagine seeing your hand rise - helpful though that perceptual type of imagining might perhaps be as an adjunct, when the agentual type of imagining is for some reason being requested of you. 

One small pebble of a stumbling block does crop up here. On the perceptual side, it is always seeing (or hearing, or feeling, or again hurting, or again in-the-passive-sense greening) that is imagined, contrary to what ordinary language in its elliptical way on occasion says. So the following forms of expression are correct only insofar as they are elliptical for the (fully correct, not elliptical) forms of autobiographical expression listed already, under the head of perceptual imaginings: 

  • "I imagine breeze-swept, waving, grass."
  • "I imagine a hand's rising." 
  • "I imagine a tiny patch of green light [or pinkish light] sitting near the top of [or descending toward the middle of] my retinal surface, as my Homo sapiens converging ocular lens projects a tiny (inverted) image." 
  • "I imagine a visual-cortex neuronal event." 


What, now, about imagining, as opposed to greening? Is there here, too, an active-passive distinction? And is there there here, too, a mindfulness-absdentmindedness distinction? I suggest - admittedly tentatively, and in a way rather more than usually subject to correction, some day, by readers - a "yes" on both counts. If this is correct, we have a list of fully four distinct possibilities:

  • actively and mindfully imagining seeing grass 
  • actively but absentmindedly imagining seeing grass (this happens to a prisoner pondering the "green, green grass of Home" in a daydreams, the moment before his evening prison reverie snaps, and he declaims in the tragic tones of Graceland, Tennessee, imitating Mr Elvis Presley, "Four grey walls surround me /.../ There's a guard and there's a sad old padre/ Arm and arm we'll walk at daybreak")
  • passively and mindfully imagining seeing grass (this happens in deliberate reverie, as when one is trying - perhaps when fighting insomnia - to relax, and one allows the imagination to drift instead of any longer directing it)
  • passively, and even absentmindedly, imagining seeing grass (as happens when the drowsiness proceeds still farther, with the backhanded, "Oh can you not just RELAX, grrrrr....", fight against insomnia now at last on the brink of success - in the last step before the blessed total swoon, between those now-so-comfortable bedsheets, under that now-so-reassuring duvet

The limits on imagination are rather severe. Just as we cannot in strict accuracy imagine grass, but only the seeing of grass, so we cannot imagine the number zero. (We can, to be sure, imagine seeing a pencil numeral, in a ring shape, on a sheet of paper. But zero is other than the particular pencil mark that may happen, as a matter of sociologically contingent notational decision, to be some particular mathematical community's current particular representation of zero. From the Middle Ages onward, we in the West have been using a ring. I think the classical Sanskrit community, cultivating mathematics in Roman times in intellectual isolation from mathematically sterile Rome, used instead something like a dot.) Zero is, rather, a thing thought about.

Further, it seems to me that we cannot even imagine imagining. The Gentle Reader has in reading this week's upload been (not imagining imagining, but rather) thinking about imagining.

Here are three small observations regarding thinking-about:

(1) Thinking-about is "supported" by (a.a) episodes of seeing (or, again, by episodes of hearing, and the like) - and additionally by (a.b) the imagining of episodes of seeing, the imagining of episodes of hearing, and so on; and again by (b.a) various kinds of doing, in hard intellectual work very typically by doings involving the marking of papers with pencils - and additionally by (b.b) the imagining of doings, sometimes notably involving the marking of papers with pencils, even blindly (as when one is walking in the park, in a worried way imagining what must now be written at the desk, even with the eyes screwed shut, once the reviving tea is brewed).

A real-life example of this, going beyond the mere banal thinking-of-zero, might help a little. It is in my amateur pedagogical judgement advisable to alert elementary-school children to at least one proposition from true Higher Mathematics - namely to the proposition, due to Georg Cantor (1845-1918), that there are multiple orders of infinity. In elementary-school terms, we develop the proposition thus: There are "more" infinite sequences of on-off binary bits (sequences in the style "0111000011111000...", with that unending-continuation-forward) than there are positive integers.

The proof proceeds through a reductio:

Suppose, per absurdum, that the infinite sequences of binary bits can be numbered by the positive integers, with not even one infinite sequence omitted from the numeration. Let the sequences, then, be exhaustively listed, with one first in the list, another second in the list, another third in the list, and so on, with say sixty per (infinitely wide, top-edged and bottom-edged and left-edged, and yet not right-edged) sheet of paper, over infinitely many sheets of paper, each with (say) the usual USA-standard  eleven inches, or the European-standard 297 millimetres, separating top edge from bottom edge. Now form the "Special Sequence of Bits", as follows: if the first listed sequence has its first bit ON, then let the first bit in the Special Sequence instead be an OFF; and if the first listed sequence has an OFF in its first place, let the first bit in the Special Sequence instead be an ON; if the second listed sequence has its second bit ON, then let the second bit in the Special Sequence instead be an OFF; and if the second listed sequence has an OFF in its second place, let the second bit in the Special Sequence instead be an ON; so so on (so that, for each k = 1, 2, 3, ... . if the kth listed sequence has its kth bit ON, then let the kth bit in the Special Sequence be an OFF, and if the kth listed sequence has its kth bit OFF, then let the kth bit in the Special Sequence be an ON). It was alleged that the listing is complete. Where, then, in this listing can the Special Sequence appear? It cannot appear first in the list, since it disagrees with the first sequence in at least its first bit. It cannot appear second in the list, since it disagrees with the second listed sequence in at least its second bit. Indeed for any k = 1, 2, 3, ... . , it cannot appear kth in the list, since it disagrees with the kth listed sequence in at least its kth bit. The supposition, then, that all the infinite bit sequences can be exhaustively listed leads to an impossibility. No: there are more such sequences than can be numbered by simply counting them off ad infinitum, as a naive-infinity "first sequence", "second sequence", ... , "kth sequence", ... . 

In reading through the explanation just given, the Gentle Reader has perhaps looked at some finite number of finite-length rows of binary bits written in pencil on paper - within some notation or other for binary arithmetic. (Nowadays, ON and OFF bits are marked with vertical strokes and rings. This is a sociologically contingent convention. I gather that at the dawn of digital computing, at Bletchley Park, the codebreaker teams instead wrote, on their wartime scratchpads with their wartime pencils, noughts and crosses  for ON and OFF - or perhaps, rather, for OFF and ON.) Perhaps, alternatively, the Gentle Reader has merely imagined seeing such things. A third possibility is that the Gentle Reader, being the kind of person who "learns by doing", has instead done some appropriate row of strokes-and-rings (or Bletchley Park crosses-and-rings?) writing - perhaps even in an Anscombean practical-knowledge spirit, with teeth gritted and eyes screwed shut in intense concentration. A fourth possibility is that the Gentle Reader might  have imagined doing such writing (while, perhaps, again in an Anscombean practical-knowledge or practical-imagination spirit refraining from imagining seeing what is written; this would be like imagining typing one's computer password while imagining looking away from screen and keyboard). In all four cases, there is some seeing, or doing, or imagining-of-seeing, or imagining-of-doing, which "supports" thinking about Cantor's topic. No matter how the thinking gets pulled off, the thought is the same - and remains the same no matter what particular binary-arithmetic notation (modern 1-and-0, wartime Bletchley Park noughts-and-crosses, or other) happens to be selected.

(2) This notion of "support" is subtle, in that the thinking of a different mathematical thought, having nothing to do with Cantor's particular theorem, could be supported by the same seeing, or the same doing, or the same imagining-of-seeing, or the same imagining-of-doing. Perhaps there is a mathematical community, somewhere, which writes little strokes and rings, or little Bletchley-Park crosses and rings, to express concepts not in set theory at all, but in classical plane geometry, or in classical spherical geometry, or in topology, or in some group-theory branch of "Abstract Algebra".

The independence of mathematical thought from mathematical-notation seeings, doings, imaginings-of-seeings, and imaginings-of-doings reminds me of a feature printed in 1950s, 1960s, or 1970s Canada by the old, and in those days entertaining, Maclean's magazine (a publication whose vagaries through the decades are chronicled at This was at a time when Canada's Good and Great were earnestly beseeching all the local Anglos, in their millions, to support bilingualism. (It was in my own private cultural judgement correct to urge this, in Canada's particular historical circumstances, quixotically divorced from real life though the urging perhaps was. It is remarkable that everyone in Anglo-Canada studies French in K-12 school, and that nevertheless only some small percentage of Anglo-Canada retains an interest in French upon graduating.) Maclean's entered into the spirit of the reigning political correctness by inviting its readers to submit poems that made sense, albeit a different kind of sense, whether read in French or in English. Astonishingly, some readers rose to the challenge, achieving some reasonable measure of success. Although I do not recall much, I do recall someone more or less successfully writing a bilingual version of a whimsical hit song of the day, "Mares [i.e., female horses] eat oats and does [i.e., female deer] eat oats and little lambs eat ivy [or, in an other version, 'little lambs eat daisies'?]" - incorporating the hauntingly faux-québecois, or faux-acadien, "Maires y d'hôtes..." - perhaps "Mayors [i.e., municipal leaders], in that place, of hostelries..." The point of this whimsical contest was that one and the same set of perception-and-action symbols could be the vehicle for two different pieces of thinking.

(3) This notion of "support" is again subtle in that thinking the normally intended thoughts, with support from some given symbolic vehicle in its sociologically common interpretation, is a different intellectual experience or intellectual act from thinking about the vehicle. One can perversely - perhaps as a designer of wallpapers - think about a finite number of finite-length rows of pencil rings and pencil strokes (and, in an Anscombean practical-intellect sense, can even perversely think about making such rows) without thinking about mathematics.

My open-ended homework is now simply this: what further things can now be said about thinking, over and above what I have said already, to differentiate it still further from perceiving, from acting, from the imagining of perceiving, and from the imagining of acting? Are there, at least, some further questions (we might not find ourselves able to give confident answers) that can be posed on this deep topic, perhaps using some notions already introduced in the various installments of this essay?

It may help some readers with my modest homework if I once again breach my Igominy and Humiligation Precept, by citing two things which have been said by philosophical authorities, and which might in at least some small way prove at least mildly suggestive.

(A) Prof. Ludwig Wittgenstein (1889-1951; for once, I may as well respectfully give his actual name) is said to have drawn attention to the special kind of thinking that accompanies the production of an indefinitely prolonged sequence of "similar things", as for example in mathematics. We ask someone to consider, as it might perhaps be, the number 2, and then the number 5, and then the number 10, and then the number 17, and then the number 26, and we ask, "Do you now see how to go on?" (What, Prof. Wittgenstein asked, is happening here? What kind of thought-performance is this, when the person declares that indeed she or he "now knows how to go on", and perhaps gratifies us by next mentioning, as we of course would in the usual "k-squared plus one" spirit hope, 37, then 50, then 65, then 82?)

(B) Prof. Peter T. Geach (1916-2013) writes, in a passage in Three Philosophers (Oxford: Blackwell, 1961: this was a collaboration with Prof. Anscombe) in which he is seeking to explain Saint Thomas Aquinas's speculations regarding the "disembodied soul": "The disembodied soul will retain the purely intelligible or logical, though not the sensuous, content of its earthly thoughts."

It will also help some readers with the homework if I invite them to consider a topic which will in any case require some mention soon, as we finish off the Geography of the Mind, namely the topic of dreaming. If you dream of falling, you are not, as a general rule, in reality falling. If you dream of something scientific - as Kekulé, for instance, has perhaps been said (whether accurately or inaccurately) to have beheld in a dream the solution to his problem regarding the shape of the benzene molecule - are you in reality thinking?

[This is the end of the current blog posting.]

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