Monday, 23 October 2017

Toomas Karmo: Part P: 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, while admittedly running behind schedule, to write out most or all of the appropriate 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. Tallinn reverts to its winter time on the last Sunday in October. Toronto revers to its winter time on the first Sunday in November. 

  • 20171025T2250Z/version 3.1.0: Kmo made some changes too numerous and tiny to invite quick summary here, but more appropriately deemed changes of substance than changes in cosmetics. (He changed a misleading "now" into the correct "not", or a misleading "not" into the correct "now"; he corrected something like "perceiving" into something like "perceiving and acting"; and so on.) He reserved the right to make nonsubstantive, purely cosmetic, tweaks over the coming 48 hours, as here-undocumented versions 3.1.1, 3.1.2, 3.1.3, ... . 
  • 20171025T2154Z/version 3.0.0: Kmo finished converting his fine-grained outline into full-sentences prose. He reserved the right to make nonsubstantive, purely cosmetic, tweaks over the coming 48 hours, as here-undocumented versions 3.0.1, 3.0.2, 3.0.3, ... .  
  • 20171025T1819Z/version 2.0.0: Kmo uploaded a properly polished fine-grained outline. He hoped to convert this into full-sentences prose by UTC=20171024T2230Z.
  • 20171024T1800Z/version 1.0.0: Kmo, already behind schedule, had time to upload merely a not-properly-polished fine-grained outline. He hoped to polish this and convert it to full-sentences prose around 26 or 28 hours from now. 

[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]

In the most recent installment of this long multi-installment essay ("Part O", from 2017-10-02 or 2017-10-03), I posed an open-ended homework question: 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? 

This week I will give part of my own effort at an answer, and will set a further homework problem to introduce the remaining part (a part which this week I leave pending). 


There seem to be at least two basic kinds of thinking. On the one hand, there is what might be called "thoughtfully being", or with equal propriety "thinking-in". On the other hand, there is what might be called "thinking about being".  

(A) Sometimes it is the case that I am undergoing something (as when I am greening-in-the-passive-sense, or am seeing grass, or am seeing a rising hand). The situation might possibly be summed up with an American Street English use of "being", in the locution "I be undergoing." Sometimes it is the case that I am doing something (as when I am raising a hand; or, again, as I would be in actively-greening, were I, counterfactually, to possess the "Active Visual Cortex" envisaged in "Part M" (2017-09-04/2017-09-05) as the special glory of some intellectually agile species other than Homo sapiens). The situation might possibly be summed up with an American Street English use of "being", in the locution "I be acting." 

All of these various specially personal ways of undergoing and acting (in the envisaged Street English, "these various specially personal ways of being") can, at least some of the time, be proceeding mindfully rather than absent-mindedly, and so can constitute ways of "thoughtfully being". All of the following, in particular, are possibilities:

  • mindfully greening, in the Estonian-passive, Mina rohetun sense (and similarly, mindfully seeing grass - what we call "looking at" grass - and mindfully seeing, i.e., looking at, a rising hand)
  • mindfully greening, in the Estonian-active Mina rohetan sense (and similarly, even for human animals, unable to glory in the envisaged Active Visual Cortex of "Part M" (2017-09-04/2017-09-05), mindfully raising a hand)
  • mindfully imagining greening, in the Estonian-passive Ma kujutan ette, et ma rohetun sense (and similarly, mindfully imagining seeing grass, and mindfully imagining seeing a rising hand)
  • mindfully imagining greening, in the Estonian active Ma kujutan ette, et ma rohetan sense (and similarly, mindfully imagining raising a hand) 
It would be equally proper in all four cases to say that one is "thinking-in". The analytical philosophers of action, the "Elizabeth Honeycombes" and the "Athur C. Dangoes" that one recalls from those old 1970s and 1980s Departments of Philosophy, rightly stressed - I have repeatedly used their helpful idea in previous installments of this essay, for my own various purposes - that one can be doing-A "in" doing-B (as when, e.g., one is entering into a contract in signing one's name, and indeed is signing one's name in moving a pen). It is now time to note that thinking itself can be "in", in broadly the same "Honeycombe-Dango" sense. Mindfully greening-passively, mindfully seeing (i.e., looking at) grass, mindfully raising a hand, mindfully imagining greening-passively, mindfully imagining greening-actively, mindfully imagining raising a hand - all such mindful undergoings and doings can properly be called instances of "thinking-in". ("Have you been thinking this afternoon? - Oh yes, I have been exercising the Perceptual Intellect in so-attentively looking at, in inspecting, in scrutinizing, the lawn"; "Have you been thinking this afternoon?  - Oh yes, I have been exercising the Practical Intellect in ever-so-attentively, in ever-so-thoughtfully, doing my prescribed hand gestures, as I rehearse for my upcoming belly-dance routine on the stage of the Blaue Engel.")

(B) Thinking about being occurs when, for instance, one reads this present essay, and therefore when

  • one thinks about greening, in the passive, Mina rohetun, sense (this can be done without actually greening-in-the-passive-sense; you the Gentle Reader, have yourself been thinking about this while looking at my essay on a computer screen, or else on some page of printout, at a time when you have surely not been directing your corneas to a Macau flag, or to a Mauritania flag, or to a sunlit lawn)
  • one thinks about greening, in the active, Mina rohetan, sense (or, again, thinks about the raising of one's hand)  
  • one thinks about the imagining of Mina-rohetun greening (this kind of thinking can be done even by humans - the congenitally colour-blind, for instance - neurologically incapable of imagining Mina-rohetun greening; "There are some things we," they truthfully think, "unfortunately cannot with our specific neurological handicap imagine - for instance, Mina-rohetun greening")
  • one thinks about the Anscombean, practical-imagination, imagining of Mina-rohetan greening (again, even congenitally colour-blind people can do this) - or, again, about the Anscombean, practical-imagination, imagining of raising one's hand
Thinking about being additionally occurs when one thinks about each of the following:

  • grass (as botany students, for example, do, in pondering chloroplasts and mitochondria: such students are not, in their acts of thinking, necessarily thinking about the  seeing of grass, or about greening in either its Mina rohetun or its Mina rohetan sense)
  • a rising hand (as anatomy students, for example do, in pondering the pull of biceps on tendon: such students are not in their acts of thinking necessarily thinking about anyone's raising a hand, though they admittedly might be) 
  • green light (as physics students, for example, might do, in thinking to themselves, "Well, the wavelength is about 500 nanometres, both in vacua and in media; and at all instants, whether or not the light is polarized, the electric and magnetic vectors are mutually perpendicular; and all the various speeds of propagation in the various transparent media are a little lower than the (invariant) speed of propagation in vacua (and so on, and so on)"
  • chirality (as solid-geometry students might do, in thinking to themselves, "The wooden R and the wooden Russian Ya (я), with their top-surface dimples [I discussed this in 'Part H', on 2017-07-17 or 2017-07-18] differ in chirality; and so do right-handed helices and left-handed helices (and so on, and so on)"
  • the number zero (as students of arithmetic might do)
In particular, the distinction between imagining passive-greening and thinking about passive-greening helps confer at least some initial mild plausibility on the late Prof. Peter Geach's aphoristic paraphrase of Saint Thomas Aquinas, as quoted here in "Part O", from 2017-10-02 or 2017-10-03: The disembodied soul will retain the purely intelligible or logical, though not the sensuous, content of its earthly thoughts. One could well envisage some thinking subjects - some angels? one or more Persons of the Trinity? some pleasantly Catholic humans, in some pleasantly respectable-Catholic state of post-death survival? - thinking coherently and insightfully about "the Toomas Karmo animal's imagining passively-greening", without themselves sinking so-to-speak down to the living Toomas Karmo level (without sinking so-to-speak so low as to be imagining anything).

To be sure, the topic of post-death survival raises questions of its own, which I will have to try to address in some small way in some later installment, once I have turned from the occasionally worrisome topics of "Perception" and "Action" to the perhaps more pervasively worrisome topic of "Subjectivity". - I do write here, "in some small way". I guess that, barring some sudden in-the-morning-shower εὕρηκα flash of insight I shall have to suggest that mere philosophy can  provide only scant consolation for would-be respectable Catholics, yearning as all of us in the respectable Catholic world admittedly are for some rigorous philosophical proof of our personal immortality.


I asked in "Part O" whether thinking occurs in dreams. In this connection, I mentioned the story of F.A. Kekulé (1829-1896), who is perhaps sometimes said to have worked out an organic-chemistry problem (specifically, the configuration of the benzene molecule) in a fit of armchair dozing. Today I answer the question in the negative, although in a provisional spirit. As always, I am anxious to be corrected by readers, should I be missing something.

The solving of chemistry (mathematics, physics, chess, ... ) problems in dreams seems to work as follows: One dreams of seeing symbols, or of writing symbols, or of seeing or doing something closely analogous, such as moving chess pieces on a board. [In general, one dreams of what in "Part O" (2017-10-02/2017-10-03) I called things "supporting" thought.] One additionally dreams of the pleasures of thinking. On occasion, one even dreams not only of seeings and doings, but in addition the imagining of seeings and doings. ("So I had this nice dream, see, in which I was walking up the Corso in Rome, wearing majorly cool Gucci shades. The October breeze was fresh on my face in the dream, and in the dream I was imagining all kinds of things - imagining seeing a most frightfully nice postcard in my right hand from Charles and Camilla, and also imagining writing down, in the margin of the postcard, with my ever-handy pencil, a most frightfully clever heat-and-work equation. And then, in this dream, I saw Don Vito Corleone approaching me on the Corso, flanked by two Roman bishops, and so in my dream I cut my imaginings short, and in my dream I knelt down to kiss the trio of approaching rings, worrying which particular ring as a matter of Sicilian and Vatican diplomatic protocol  ought to come first, and then in my dream Don Coreleone said to me ...") Upon waking, one recalls what it was one dreamed, and now one realizes that the dream content - amazingly, since dream content is for the most part merely confused - contains a solution to a physics problem. (Maybe the heat-and-work equation, in this example not even a thing dreamed of being seen, or dreamed of being written,  but a thing dreamed-of-being-imgined-to-be-seen or dreamed-of-being-imagined-to-be-written, really was clever. Some maths or physics students might be luckier in this regard than I generally, as an undisciplined dreamer, have managed to be.) The thinking occurs not during the dreaming, but afterward, when one examines the dream content, even as one might examine an incoming e-mail from some helpful university tutor. 

To be sure, there is also such a thing as the "Lucid Dream" ( We dream "lucidly" when we are aware of dreaming, and quite mindfully construct (to take one example) witty rebukes to people who in a dream have been pressuring us to buy frivolous merchandise - well aware that the whole thing is a dream, and that the outright opening of eyelids, to face the duties of the morning, cannot now be far off. In this special case, I hesitantly suggest, we do have the coinciding of dream state with thoughtfully-being (if we are mindfully really selecting, with unreal dream-larynx, the words of a witty rebuke) or with thinking-about-being (if we are, in the manner of the alleged Kekulé, mindfully solving the Configuration-of-Benzene problem).

A nagging worry arises with special force regarding dreams in which one is addressing some practical problem. As I dream of wondering which ring to kiss first, in that ominous Corso meet-up, am I not thinking while dreaming? I have made up all my dream examples in this essay. But we will all admit that in our actual dreams - as we might write them down for, perhaps, some investigator, some day, in some Toronto or Tallinn psychology lab -  there seems to be a lot of pondering of practical problems. It is common enough to dream of being pursued, and of seeking an appropriate avenue of escape. I am unsure of my ground here. Maybe there is no real thinking when the dream presents an urgent practical conundrum, but a mere dreaming of the supports of thinking. I am more inclined, however, to concede that there is on at least some occasions already a modest kind of thinking here, in which the mind is active even as in the more dramatic case of Lucid Dreaming. It is as though we can think when moving real chess pieces on a real board, in waking life, and also can dream in bed, thinking real thoughts about what we erroneously take to be a real Mafia Don and a pair of real Prelates. So maybe, after all, we can solve the Benzene Configuaration Problem by thinking while dreaming - though I am still inclined to suggest that the more usual thing is a mere dreaming of the symbolic supports of the thinking, with their true import being in the most usual cases thought through upon waking.

Wittgenstein is famous for having discussed the seeing of an ambiguous drawing "either as a duck or as a rabbit" (, subsection headed "Seeing that vs. seeing as"). I suggest, again hesitantly and tentatively, that seeing those heavy black lines as (say) a mallard duck is one kind of thinking-in, somehow bound up with thinking-about. One is thinking-of-being (specifically, one is thinking of one particular kind of avian being, ultimately involving Anas platyrhynchos in the family Anatidae) in the seeing of the lines. It perhaps counts in favour of my hesitant suggestion that if one is seeing the lines in a thoroughly absent-minded spirit, in other words is seeing-without-at-all-looking, then one will be neither seeing them as a mallard nor seeing them as a rabbit. Similarly, I suppose, if one is imagining the lines in an absent-minded reverie (as discussed in the context of Mr Elvis Presley, back in "Part O"), then one can be neither imagining them as a mallard nor imaging them as a rabbit. (No: if either being-a-mallard or being-a-rabbit is in the mind, then some thinking is going on, with the imagining no longer absent-minded.)

What, now, about dreaming of seeing the Wittgenstein drawing?  Can one not dream of seeing the lines as a mallard, and of a moment later of seeing the lines as a rabbit? Surely this is a possible kind of dreaming, although in actual psychology-of-sleep laboratories rare. So, I suppose (subject to correction by others) that here, just as in the case of thinking-under-the-duvet-about-Sicilian-and-Vatican-diplomatic-protocol, and even on some conceivable rare occasions about the benzene configuration, there is a modest kind of actual thinking.

And how does dreaming relate to hallucinating? Here I am at a loss on what to write.


The reader who has thought about the homework which I set at the end of "Part O" will now have posed for the philosophically troubling mental feat of thinking a pair of questions drawing on notions I have already used for the (philosophically less alarming?) mental feats which are perception and action:

  1. Is there a possible distinction between thinking "absentmindedly"and thinking "mindfully" - as I have suggested that we can be (a) absentmindedly seeing grass or mindfully seeing (looking at) grass, (b) absentmindedly raising a hand or mindfully raising a hand, (c) absentmindedly imagining the seeing of something or mindfully imagining the seeing of that thing ("Well, I am imagining that notorious line drawing, and I am saying to myself at this moment, MALLARD! MALLARD! - and now, instead, RABBIT! RABBIT!"), and (d) absentmindedly imagining doing something or mindfully imagining doing that thing? 
  2. Is there a possible distinction between thinking "actively" and thinking "passively" - a distinction, that is, mirroring the distinction between actively and passively "greening", and again the distinction between actively imagining (as in anxious insomnia) and passively imagining (as at the blissful edge of approaching sleep)? 
I tentatively suggest a "no" in both cases.

  1. As far as I can see, the notion of absentmindedly thinking-about-being is a contradiction in terms. The closest we can come to this is absentmindedly imagining, whether actively or passively, some pencil writing some helpful thermodynamics equation, or one's writing an equation. (Here are two things to be imagined - one Perceptual, one Agentual - with each to be imagined either actively or passively. So we have here fully four distinct ways of being absentminded.) Such imagining is an imagining of the mere "supports" of thinking. If thermodynamics thinking ever does occur, it is not in that act of imagining, but in a subsequent act of recalling and pondering - in a mental feat akin to the recalling and pondering of some diagram once seen on the blackboard, or of some words once heard on the radio. - Still more is it a contradiction in terms to suppose someone to be absentmindedly "thoughtfully being". If you are thoughtfully, say, raising an arm, perhaps while rehearsing for the Blaue Engel stage show, then you are doing so mindfully - and yet the performance of mindfully-raising-an-arm cannot itself be mindless. (That is to say, one cannot be mindlessly mindfully-raising-an-arm. "Mindfully" and "mindlessly" are in this respect a different kind of adverbial-contraries pair from the contraries "right-handedly and "left-handedly", as say, applied to a spring of 1-millimetre coil radius coiled up loosely in a wide-mouthed jar of radius 5 centimetres, for convenient storage. Such a spring could be right-handedly left-handedly coiling - as one should point out, with an actual left-handed spring coiled up in lazy right-handed turns in a big glass jar,  to K12 students, when being duly thorough in teaching them chirality.)
  2. There is, to be sure, such a thing as "compulsive ideation". This thing, however, occurs at the level of imagining, and not at the level of thoughtfully being or thoughtfully thinking about being. Let me try here to envisage, extrapolating in extravagant drama-queen style from my own modest private experiences, which it might mean to be an actual hospital patient, mentally ill with actual intergenerational PTSD, and specifically through "being obsessed with the Camps". In what would that consist? Simply in one's imagining, compulsively, - and for the most part passively, for the most part no doubt wishing not to be enduring such intrusive imagery - the cattle cars, the piles of cadavers, the ovens, day upon sombre day. The imaginings are not themselves thinkings. They are at best supports for episodes of thinking. The nurse may well say, "This psychiatric patient is passive in respect of his imaginings," without thereby touching on the patient's thoughts. For all the nurse has said, it might be that the patient, while perpetually imagining the Camps, is at some instants in his sombre day (a) thinking about the Camps (as in pondering, in the footsteps of professional philosophers-of-law and professional ethicists, the correctness or incorrectness of the 1945 "Collective Guilt" suggestion), at other instants (b) being-thoughtfully (being in a particular imaginative state, and mindfully being in it) while not thinking-about-being, and at yet other instants (c) neither thinking-about-being nor even being-thoughtfully. (Instants in the third category are instants at which the envisaged psychiatric patient is in sheer - no doubt in fortunate - absentmindedness either performing or enduring the imagining, perhaps while now pottering about, calmly enough, in the hospital's Occupational Therapy rooms.) 
In suggesting a negative answer to the second question, I am rejecting the idea that thoughts (as distinct from imaginings) "might on occasion be forced into one's head".


It is appropriate to finish by setting up a piece of homework. What more can be said on the topic of "Thinking about Being"? In the last installment, I mentioned thinking about chirality and thinking about zero. Consider now a third example of mathematical thinking, namely thinking about randomness and nonrandomness. Let the Gentle Reader consider what is meant by claiming an infinite sequence of bits - say, a sequence that begins 10100010000001011011101100101110... - to be "random", or again to be "not random". We in some sense "rationally expect, without asserting certainty" that in a truly random sequence, any finite segment that is "sufficiently long" will contain "approximately equal numbers of ON bits and OFF bits". But what is meant here by the words "rationally expect" and "sufficiently" and "approximately"? Is it at all possible to give a rigorous definition of "random" and "nonrandom", free of such cheap weasel words?

O Gentle reader: Is the concept of a Turing Machine, with its infinite, initially blank, read-write tape and its program of finite length, helpful in the quest for rigour? The challenge is to (a) try, as one's best effort, to write definitions of the concept of "random sequence'" and "nonrandom sequence" which somehow incorporate such phrases as "There exists at least one Turing Machine such that zacka-zacka," or again "Every Turing Machine which is such that zecka-zecka is such that zicka-zicka," and to (b) write out some critique of the proffered best-effort definitions - a critique perhaps sympathetic, perhaps unsympathetic, and perhaps comprising even a mixture of the sympathetic and the unsympathetic.

In writing out the critique, the Gentle Reader should consider also the question, "On my definition, does it become a known fact that somewhere in the vast universe of infinite bit-sequences, some truly random sequences do exist? Or is this, rather, a point on which, under my best-efforts definition, humanity is doomed to agnosticism?"

And finally, the Gentle Reader should try to relate this example to the notion of thinking-about-being, perhaps seeking with the aid of this troubling, concept-of-randomness/concept-of-nonrandomness example, to show that the notion features hard-to-plumb depths.  

In providing a solution for the homework - I hope, in the next installment - I rather expect to be referring to the Wittgenstein "now-I-know-how-to-go-on" infinite-sequence scenario with which I ended "Part O".

Some readers may be helped in the homework by the following minor remarks, with which I shall have to end this week:

  • In practical engineering, it is required that people should have available to them random bit sequences, or at any rate sequences that in some (philosophically troubling!) sense manage to be "close enough to being random, without being truly random". Here is a practical illustration. We are in a branch of government, procuring many tens of thousands of halogen light bulbs as a shipment from some corporation. We write into the contract a stipulation  regarding quality - "In a random sampling of 700 bulbs in our own civil-service workshops, at least 99.9% of the bulbs in the shipment shall be found to achieve a lifetime exceeding 500 hours, when run by us at a voltage 50% higher than the vendor's advertised power-mains rating." We, as government officials, now set aside the three weeks needed for that 500-hour test, selecting 700 bulbs "at random" from the big ocean-shipping container. To our dismay, we get a poor outcome, with just 92.1% of the tested bulbs lasting the full 500 hours. So we send that ocean-shipping container, with its many tens of thousands of untested, but now untrusted, bubs back, and we decline to pay the invoice. To our further dismay, the corporation is not only disagreeable, but even impertinent, going so far as to sue us. What happens in court? Some lawyer defends the corporation, challenging us to prove that we selected our disappointing bulbs, as it was written in our mutually binding contract, "at random". Well, we reply, we in the Government of Canada (or in the Government of Estonia, or whatever) did not do anything so crude, and so open to legal challenge, as to draw the bulbs at whim, with our own itchy little fingers. No Ma'am, no, Your Honour: we were correctly rigorous, as per the contract. We asked a government statistician to supply us with some finite-length initial segment of a random bit sequence. We drew the bulbs for testing not at our whim (we concede to the Court that a whim, however innocent, can harbour unconscious bias), but under the guidance of the bit-sequence our official statistician had written for us. (We perhaps took the first few thousand bits and chopped them into octets, as when 10100010000001011011101100101110...gets chopped into 10100010,  00000101, 10111011, 00101110, ... . Each octet is the binary representation of some integer in the range 0, 1, 2, ..., 256. So now we had lots and lots of random integers, in convenient decimal notation on our clipboards at the shipping dock. We thereupon used our clipboards to work out which pallets to unload, and from which pallets to extract which carton(s), and from which carton(s) to pull out which particular bulb(s) for testing.) So we explain, in that scary courtroom. And now the scary corporate lawyer asks us, "How do you know your test-regulating sequence to have been truly random?" On our ability to answer the lawyer's comeback question now turns our legal case, with perhaps many hundred thousand dollars of taxpayers' money (or many hundred thousand Euros of taxpayers' money, or whatever) now at stake. 
  • The practical engineering task is addressed from Switzerland by an American thinker, Mr or Dr John Walker, who earlier in his life was instrumental in the development of CAD-CAM ( This authority, mindful of the need for random numbers in industry and government, offers a random-number server at One might in the same spirit cite a further example, the Australian National University's random-number-server offering at
  • Turing Machines are conceptually less scary than they perhaps sound. The topic is (a) readily researched through Google, and (b) already appropriate for inclusion in the humble K12 curriculum, let alone in such things as third-year university logic. 
  • In pondering Turing Machines and randomness, one can usefully ask, in the spirit of my remarks from "Part O", "The (clearly infinite) collection of possible finite-program-length, infinite-initially-blank-tape, Turing Machines is an infinite collection of what order of infinity? Have we here a collection which is infinite in the modest sense in which the positive integers are an infinite collection? Or have we instead a collection infinite in, say, the sense [as noted in 'Part O', less modest] in which the entire collection of infinite bit-strings is infinite?" 

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



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