Let’s start with lists.

Lists are very important in mathematics — not because lists are especially interesting as a concept, but because they are a basic structure for other, more interesting things.

We start with some examples of lists and introduce some vocabulary.

1. [2,3,5,7,11,13,17]
2. [2]
3. [2,2,2,2,2]
4. [red, green, blue]
5. [ ]
6. [ [ ], 2, red ]

The first list is an ordered list of numbers. There are seven items in this list, so we say that the length of the list is seven. The first item in the list, sometimes called the head item of the list (or simply the head of the list) is the number 2.

The second list contains only a single item (its head) and is this called a singleton list. Its length is one and its head is the number 2.

The third list consists of five copies of the number 2. It has length five and is headed by the number 2. Because all the items in this list are constantly the same, we say this is a constant list.

The fourth list is a list of three colours. Its length is three and its head is the colour red.

The fifth list is a very special list called the empty list. Its length is zero and it contains no items. Since it has no items, there is no first item. The empty list is the only list that has no head.

The sixth is a list of length three. The first item in the list is itself a list (the empty list). Its second item is the number 2. It’s third item is the colour red. Because the items in this list are different kinds of things (list, colour, and number) we call this a heterogenous list (hetero = different), unlike the other lists which are homogenous lists (homo = same).

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We can join two lists together to make a new list in an obvious way. We call this list catenation. We will use the infix operator ‘++’ to denote the catenation operation. This is best introduced by examples:

1. [0,1] ++ [2,3] -> [0,1,2,3]
2. [2,3] ++ [0,1] -> [2,3,0,1]
3. [ ] ++ [red, green] -> [red, green]
4. [red, green] ++ [ ] -> [red, red]
5. ( [0] ++ [1] ) ++ [2] -> [0,1] ++ [2] -> [0,1,2]
6. [0] ++ ( [1] ++ [2] ) -> [0,1,2]

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These examples illustrate the MONOIDAL properties of list catenation. Examples (3) and (4) illustrate that the empty list serves as a NEUTRAL element for catenation, while examples (5) and (6) illustrate the ASSOCIATIVITY of catenation.

Note that the catenation operation is NOT COMMUTATIVE. This is illustrated by examples (1) and (2). But I will give another example:

1. [0,1] ++ [0,2] -> [0,1,0,2], but
2. [0,2] ++ [0,1] -> [0,2,0,1]

Swapping the left and right operands [0,1] and [0,2] clearly yield different results, because the resulting lists [0,1,0,2] and [0,2,0,1] are not the same. Right? RIGHT?

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FINALLY, SOME META QUESTIONS TO PONDER..

• Are the lists [0,1,0,2] and [0,2,0,1] really not the same?
• Is it clear that they are not the same? (You might ask in response: clear to whom?)
• If is clear THAT they are not the same, is it clear WHY they are not the same? (You might again ask in response: clear to whom?)

The purpose of this introduction to lists was (in part) to set up these meta questions, and the purpose of these meta questions is to motivate our next topic: the difference between intuitive thinking and rigourous thinking in mathematics and programming.

The Open Boat

This is the title of a short story written by Stephen Crane. In this story, Crane describes (in a fictionalized way) his recent experience. It was 1897, and Crane was sailing from Florida to Cuba to work as a newspaper reporter. En route, the boat was shipwrecked. Crane, with three other men, spent thirty hours rowing a lifeboat in high seas before reaching shore.

The story is a classic of American literature. It’s themes of hardship, of endurance, and of the bond formed between men in times of adversity, make it an excellent read for any young man who has some knowledge of English as a Second Language, and who wishes to develop his vocabulary and his ear for style.

There is a nice narration of party of the story, with graphics and explanation, on youtube at the following link:

https://www.youtube.com/watch?v=ipKXvkFTEMc

The full text of the story, nicely formatted, can be found at the following link:

https://americanenglish.state.gov/files/ae/resource_files/the-open-boat.pdf

A perfect lesson.

The twenty-two minute video [Monoids] on the YouTube channel [All Angles] is perfect lesson. It introduces the viewer to what, for many, is a mathematical structure that they have never heard of, and which is typically first introduced in an upper level university math course. But after viewing this video, many may be left wondering why they had not been taught this in primary school.

It introduces the abstract notion of a monoid through animated visual examples. The narration never distracts from the visual presentation, but is an ongoing commentary on precisely what the viewer is seeing. The pace is perfect, never rushed and never dragging.

On Angst

(A lengthy youtube comment from seven months ago.
See https://www.youtube.com/watch?v=JBAhnxmU7o4 )

Are the existential ‘problems’ rightly understood as Wittgensteinian pseudoproblems?

There is meaning in the world. At least, there is purpose in the world. Biology would be incomprehensible without recognizing purpose. So we can dismiss claims that the world is meaningless or that it is without purpose simpliciter.

This being so obvious, an angsty existentialist (forgive the portrait please!) must resort to linguistic qualifications by saying things like ‘there is no ultimate purpose’. But we should ask our angsty friend what she, he, or it, means by ‘ultimate’ and, crucially, what function that word plays in our language games?

Our angsty friend will likely find that he can find no language games that arise out of our normal forms of life. It is useful to speak in terms of (say) economic purposes, or linguistic purposes, or biological purposes. Such talk is meaningful because such talk is itself purposeful in addressing different aspects of life in the world. It is in the utility of such speech that the speech has meaning.Meaningful speech is, to Wittgenstein, speech that serves a purpose.

Here we must recognize that if someone is seeking angst, one cab employ the phrases like ‘ultimate purpose’ as a quasi-entity, for the purpose of denying this entity’s existence and so finding a cause for lament. But this does not mean the phrase ‘ultimate purpose’, as used by the angsty existentialist, has any ontological or moral significance outside if the contrived purpose of seeking angst, in which case we can speak of the phrase’s psychological significance, but it can be doubted whether this significance is healthy. If people are reifying phrases like ‘ultimate purpose’ for the purpose of anxiety, this is perverse. If people are reifying such phrases because the vagaries of grammar led their metaphysics astray into thinking it is something to be denied on metaphysical grounds, and that the angst is an unsought byproduct, then recognizing that language of ‘ultimate purpose’ (and the like) is, actually, intellectual non-sense, then the angst can be of, and future intellectual effort redirected.

If, on the other hand, our angst existentialist insists that phrases like ‘ultimate purpose’, as she, he or it uses them, has intellectual coherence that entail ontological and moral significance, (and note that their use of such terns tacitly presumes such coherence, for such coherence is a prerequisite to arguing for such significance), then I invite her, him, or it, to expound on the meaning of the phrase ‘ultimate purpose’ in a way that lets engage their implicit argument.

For their implicit argument is, by an large, to a moral and/or psychological (or, as they might calk their malange of the two, ‘existential’) crises. I ask: whence the crises?

If by no ‘ultimate purpose’ one means simply that one believes (for sound scientific reasons) that, come the heat death of the universe, purposeful activity will have ceased (the word ‘ultimate’ meaning ultimate (final) in a temporal sense), then I might agree. But that does not mean there is no actual purpose now. I don’t despair of talk of purpose, even with regard to my life, just because of that bare fact.

The angst in the conclusions drawn suggests to me that the term ‘ultimate’ is carrying more weight than the bare fact of heat death. Why is heat death in the future a cause of exisential angst? Is this angst the result of making the long term future of the universe a psychological idol, and the angst a result of the idol being threatened? If so, then such existential crises is interesting as a form of psycholgical pathology. It is of psychological and medical interest. But not of philosophical interest.

If, however, a communicable case can be made that this heat death constitutes a foundation for a current crises [what sort of crises? existential crises. what is _that? ], then lets hear it. It might be philosophically interesting. Or if such crises properly follows some less remote scientifically grounded premise (e.g. that you will someday die), then lets here that.

I suspect that the what makes these things actually problematic to people is nor that they constitute a genuine philosophical problem. I suspect that they constitute a heart problem arising from disordered attachments of the heart. If so, the phenomena remain interedtimg, just not phiisophically interesting (outside of perhaps the applied field of philosophical anthropology. )

Any pushback from thoughtful existentialists would be welcome. Philosophically speaking, are existential problems what Wittgenstein would call pseudo-problems (at the level of theory, for they may still be psychologically problematic for the disordered), problems which on the philosophical level are not to be dis-solved, perhaps dissolved into tractable problems for special sciences like psychology? Or are the problems genuinely philosophical in a way not merely ‘dissolvable’ my Wittgenstenian language analysis? Cheers!

My new favorite math problem!

Given: One hundred men are sentenced to be hanged. Their warden decides to give them a sporting chance at being pardoned. He says to them: “Each man has their photograph lying face down on a table in a nearby room. Each man is invited to enter the room alone and to overturn half the photographs. If a man overturns his own photograph, he gets a gold star. Each man must leave the room as he found it. If every man gets a gold star then all will be pardoned; otherwise all will hang. You may confer among yourselves.”

Question: Were the men given a truly _sporting_ chance?

Notes: No trick language involved. Natural assumptions (one name per box, etc) apply. The only strictly non-mathematics part of the puzzle is what constitutes a _sporting_ chance. I will leave that up to the problem solver.

Hint: I have seen this presented with the men instead being numbered ‘1’ to ‘100’, with boxes containing each man’s number instead of photographs showing each man’s face, and (hint hint!) with the boxes themselves labeled ‘1’ to ‘100’. Enjoy!

OMQ

The VOCAB file can be found here : https://docs.google.com/document/d/1KBvTpsVwQ_y-O2gRSTaMIDTTngzVO47f/edit?usp=sharing&ouid=109767092901987067278&rtpof=true&sd=true

The NOTES file can be found here: https://docs.google.com/document/d/1n_bcqq-5o8kUyc0ZMKFZvBViY1gtrfU_/edit?usp=sharing&ouid=109767092901987067278&rtpof=true&sd=true

The Old Master Q comic strip offers rich material for practicing and developing English composition. I here outline a sample lesson that used one of these strips.

Writing involves two things: deciding what to write about, and deciding how to write about it. We here tell (or more accurately show) our writing students what to write about, allowing them to practice the how without distraction. We do this by providing the student with an OMQ comic (usually in an image file called COMIC) and ask them to write a description of what it shows.

We guide them first by providing a list of appropriate vocabulary items (=words and phrases) in a file called VOCAB. The student’s first goal should be to become familiar with the VOCAB terms. We help them by providing inline notes to these terms. These notes appear as as bullet points under their terms in the VOCAB file. These notes may range from definitions and use cases to etymological and morphological analysis. They are meant to help students get a head start on guessing the basic meaning of the VOCAB terms.

Secondly, we provide a list of extra notes in the NOTES file. These extra notes will give the student a deeper understanding of vocabulary terms. They will also hint at how these terms might be used in describing the comic. The student is expected to read these extra notes and make them inline notes by determining which VOCAB item(s) the extra notes belong to, and then copying them appropriately to the VOCAB file. Doing this will make the student focus on, and think about the meanings of, these extra notes. This will help the student to internalize the meanings.

The student should now be able to read the comic and identify panel(s) where the vocabulary item is useful. After each item in VOCAB, the student should indicate which panel(s) have scenes which can use the item. Now the student should begin to write a description of the comic using appropriate words and phrases. (Not every VOCAB term needs to be used, and the student can use terms not found in VOCAB. But the student should consider VOCAB to be a good model.)

This description may contain many short simple sentences. This is fine for a first draft. After finishing a first draft the student should ‘take a break’ and ‘sleep on it’. After ‘sleeping on it’ the student should revise their first draft. When they revise they should try to combine the simple sentences into more complex sentence — especially sentences with subordinate clauses. There will me many ways of doing this. The student experiment. Note that very complicated sentences can be hard to read and are often not good style. (Sometimes, simpler is better.) But it is good practice for an ESL student to try writing some very complex sentences (with subordinate clauses containing subordinate clauses). Doing this will exercise and strengthen their understanding of grammar.

The files COMIC, VOCAB, and NOTES will be provided to the student a week before their lesson. The student should aim to provide the teacher with a revised draft before each lesson. During each lesson the teacher will help the student with their descriptions. The student should at least have a substantial first draft for the teacher to look at before the beginning of the lesson.

“Deacon Blues” by Steely Dan.

The song “Deacon Blues” by Steely Dan has it’s titular character narrate what seems to be his story, but this song is less about telling a story than it is about painting a portrait of someone narrating what he *wants* be his story. What follows are a few (very subjective) notes on how this relates to the song’s use of grammatical tense.

The narrator at the start of the song is milquetoast. He has been an observer of ‘life’ (i.e. the sort of life he longs to live). He tells us that, at long last, he is about to live it. This life (or lifestyle) is one of idealized freedom and self-destruction. I claim that the song is about just this: his idealization of this lifestyle, and his (if we accept him at his word) readiness to live it. The song does not tell a story but rather paints a portrait of a man who narrates what is not so much a story of himself as a description of a lifestyle he longs to live. What story is given is threadbare: he will become a musician, he will drink and gamble, and he will die behind the wheel — less a story of a man than a portrait of a lifestyle that he has long desired, and will now begin to live.

An objection to this claim is that the narrator sometimes uses the present and past tense to describe what he is doing or has done. This has led at least one person to describe the song as the story of a musician describing what life on the road is like. It is this description that prompted me to write this short analysis. So let’s address the tense issue.

Although he does use the present tense when he speaks of his bittersweet experiences banging languid women, the chorus (in all but its final version) keep bringing the listener back to a condition of anticipation ( ‘I /will/ learn to work the saxophone, I /will/ play just what I feel’). Unless we think the writer (whom we will presume is competent) wants us to feel a kind of temporal whiplash upon return of the chorus, we should hear the narrator as speaking in an kind of /anticipatory present tense/ when talking about the ‘languid and bittersweet’. I.e. the narrator, while having not yet having had such experience, is speaking from the perspective of one who has.

More important than this, though, is what it means for character. This is a song whose purpose is, first and foremost, to paint a portrait of a man gripped by romantic idealism. This idealism is subverted to the degree that he is taken to be describing his actual lived experience. The narrator is portrayed from the start as man who has often thought of, but has until this point, lacked the courage to ‘cross that fine line’, and this is a portrayal that the chorus continues to refresh. This portrayal of naive commitment to a imagined and long desired way of life is undermined if we read him anywhere as speaking from actual experience.

The final version of the chorus does put everything in the past tense, and it is satisfying, because, by means of the past tense, it /underwrites/ the fantasy. Although one can interpret the narrator here as speaking in the /anticipatory past tense/ one can hear it as an actual summation (from beyond the grave) of the fulfilment of the fantasy. But it affirms the elements of the fantasy only. It lacks particularity. And by being a reflection of the past, rather than an in situ description, it comes across less a veracious account than an idealization of what has preceded. Whether this is an an anticipatory idealization or actually a retrospective idealization matters little. What is maintained a portrait of one who, at least at the start, was milquetoast, and whose description of the lifestyle desired (or lived) is idealistic.

What is painted is a picture of an idealized debauched and self-destructive lifestyle through the eyes of a poor sap disillusioned with his current life, in inimitable Steely Dan fashion.

Do we have to choose just one interpretation of that final chorus when listening? Not at all. Good Scotch whisky does not have to be single malt. I listen to this song with a sense of the ‘reality’ of this song being a blend of the two. And the beauty of songs with such ambiguity is that you can blend it to your own taste.

I’ve been banned from Facebook

I’ve been banned from facebook for seven days because I posted ‘Hitler did nothing wrong while I was alive’.

I was born in 1965, about twenty years after died.

Now I have important business on hold. This capriciousness is not good.

Understanding Our World: An Invitation

I would like to invite you all to join me in a disquisition. What is a disquisition? It is a kind of journey of the mind. One might call it an adventure. It means a discussion of a topic that can be as long and as elaborate as any other adventure — as narrow as a search for a missing treasure or as broad as the survey of a new world.

I have given this disquisition the tile Understanding Our World for two reasons. First it is homage to the aptly book ‘Understanding Our World’ by Hendrik Hart which has helped me to understand our world, with my own understanding. Second, the object of the book and the object of my disquisition are much the same.

And now for something completely different: Jordan B Peterson and the Bible

This post brings to a close last year’s “All Wittgenstein all the time” focus of this blog. I will experiment with it being a place for sharing resources with others (instead of merely a place for me to develop my own thoughts.)

Expect to see a lot of Jordan Peterson material posted here. This series of his which he titled “The Psychological Significance of the Bible” is *not* the best way of finding out quickly what Peterson is about. The entire series is a daunting 30+ hours long, and growing. But there are some who want to explore it. At the time of posting, I am into the fourth of the series.