As per my declaration I’d do these reviews when I had about seven to ten comics to show off, I’m entering another in the string of mathematics-touching comic strip summaries. Unless the last two days of the year are a bumper crop this finishes out 2012 in the comics and I hope to see everyone in the new year.
I’ve been trying to balance how often I do the comics reviews with how often I do other essays; I admit the comics feel like particular fun to write, but the other essays are less reactive. This leaves me feeling like after I’ve done a comics roundup I should do a couple in which I come up with the topic, the exposition, and all the supplementary matter for a while, but that encourages pileups in the comics. I’m thinking of shifting over to some kind of rule less dependent on my feeling, such as writing a comics article whenever I have (say) seven to ten features to show off. We’ve got that more than that now, it turns out, so let me start out with some that came across my desktop since the last comics review.
If you haven’t seen the Aardman Animation movie Arthur Christmas, first, shame on you as it’s quite fun. But also you may wish to think carefully before reading this entry, and a few I project to follow, as it takes one plot point from the film which I think has some interesting mathematical implications, reaching ultimately to the fate of the universe, if I can get a good running start. But I can’t address the question without spoiling a suspense hook, so please do consider that. And watch the film; it’s a grand one about the Santa family.
The premise — without spoiling more than the commercials did — starts with Arthur, son of the current Santa, and Grand-Santa, father of the current fellow, and a linguistic construct which perfectly fills a niche I hadn’t realized was previously vacant, going off on their own to deliver a gift accidentally not delivered to one kid. To do this they take the old sleigh, as pulled by the reindeer, and they’re off over the waters when something happens and there I cut for spoilers.
I’m sorry to have fallen quiet for so long; the week has been a busy one and I haven’t been able to write as much as I want. I did want to point everyone to Geoffrey Brent’s elegant solution of my puzzle about loose change, and whether one could have different types of coin without changing the total number of value of those coins. It’s a wonderful proof and one I can’t see a way to improve on, including an argument for the smallest number of coins that allow this ambiguity. I want to give it some attention.
The proof that there is some ambiguous change amount is a neat sort known as an existence proof, which you likely made it through mathematics class without seeing. In an existence proof one doesn’t particularly care whether one finds a solution to the problem, but instead bothers trying to show whether a solution exists. In mathematics classes for people who aren’t becoming majors, the existence of a solution is nearly guaranteed, except when a problem is poorly proofread (I recall accidentally forcing an introduction-to-multivariable-calculus class to step into elliptic integrals, one of the most viciously difficult fields you can step into without requiring grad school backgrounds), or when the instructor wants to see whether people are just plugging numbers into formulas without understanding them. (I mean the formulas, although the numbers can be a bit iffy too.) (Spoiler alert: they have no idea what the formulas are for, but using them seems to make the instructor happy.)
My Dearly Beloved, a professional philosopher, had explained to me once a fine point in the theory of just what it means to know something. I wouldn’t presume to try explaining that point (though I think I have it), but a core part of it is the thought experiment of remembering having put some change — we used a dime and a nickel — in your pocket, and finding later that you did have that same amount of money although not necessarily the same change — say, that you had three nickels instead.
That spun off a cute little side question that I’ll give to any needy recreational mathematician. It’s easy to imagine this problem where you remember having 15 cents in your pocket, and you do indeed have them, but you have a different number of coins from what you remember: three nickels instead of a dime and a nickel. Or you could remember having two coins, and indeed have two, but you have a different amount from what you remember: two dimes instead of a dime and a nickel.
Is it possible to remember correctly both the total number of coins you have, and the total value of those coins, while being mistaken about the number of each type? That is, could you remember rightly you have six coins and how much they add up to, but have the count of pennies, nickels, dimes, and quarters wrong? (In the United States there are also 50-cent and dollar coins minted, but they’re novelties and can be pretty much ignored. It’s all 1, 5, 10, and 25-cent pieces.) And can you prove it?
A book I’d read about the history of New Jersey mentioned something usable for a real-world-based problem in fraction manipulation, for a class which was trying to get students back up to speed on arithmetic on their way into algebra. It required some setup to be usable, though. The point is a property sale from the 17th century, from George Hutcheson to Anthony Woodhouse, transferring “1/32 of 3/90 of 90/100 shares” of land in the province of West Jersey. There were a hundred shares in the province, so, the natural question to build is: how much land was transferred?
The obvious question, to people who failed to pay attention to John T Cunningham’s This Is New Jersey in fourth grade, or who spent fourth grade not in New Jersey, or who didn’t encounter that one Isaac Asimov puzzle mystery (I won’t say which lest it spoil you), is: what’s West Jersey? That takes some historical context.
It’s been long enough since my last roundup of mathematics-themed comics to host a new one. I’m also getting stirred to try tracking how many of these turn up per day, because they certainly feel like they run in a feast-or-famine pattern. There’d be no point to it, besides satisfying my vague feelings that everything can be tracked, but there’s data laying there all ready to be measured, isn’t there?
fluffy, one of my friends and regular readers, got to discussing with me a couple of limit problems, particularly, ones that seemed to be solved through L’Hopital’s Rule and then ran across some that don’t call for that tool of Freshman Calculus which you maybe remember. It’s the thing about limits of zero divided by zero, or infinity divided by infinity. (It can also be applied to a couple of other “indeterminate forms”; I remember when I took this level calculus the teacher explaining there were seven such forms. Without looking them up, I think they’re but I would not recommend trusting my memory in favor of actually studying for your test.)
Anyway, fluffy put forth two cute little puzzles that I had immediate responses for, and then started getting plagued by doubts about, so I thought I’d put them out here for people who want the recreation. They’re both about taking the limit at zero of fractions, specifically:
where e here is the base of the natural logarithm, that is, that number just a little high of 2.71828 that mathematicians find so interesting even though it isn’t pi.
The limit is, if you want to be exact, a subtly and carefully defined idea that took centuries of really bright work to explain. But the first really good feeling that I really got for it is to imagine a function evaluated at the points near but not exactly at the target point — in the limits here, where x equals zero — and to see, if you keep evaluating x very near zero, are the values of your expression very near something? If it does, that thing the expression gets near is probably the limit at that point.
So, yes, you can plug in values of x like 0.1 and 0.01 and 0.0001 and so on into and and get a feeling for what the limit probably is. Saying what it definitely is takes a little more work.
So. The really big flaw in my analysis of an “Infinite Jukebox” tune — one in which the song is free to jump between two points, with a probability of of jumping from the one-minute mark to the two-minute mark, and an equal likelihood of jumping from the two-minute mark to the one-minute mark — and my conclusion that, on average, the song would lose a minute just as often as it gained one and so we could expect the song to be just as long as the original, is that I made allowance for only the one jump. The three-minute song with two points at which it could jump, which I used for the model, can play straight through with no cuts or jumps (three minutes long), or it can play jumping from the one-minute to the two-minute mark (a two minute version), or it can play from the start to the second minute, jump back to the first, and continue to the end (a four minute version). But if you play any song on the Infinite Jukebox you see that more can happen.
Working through my circle of friends have been links to The Infinite Jukebox, an amusing web site which takes a song, analyzes points at which clean edits can be made, and then randomly jumps through them so that the song just never ends. The idea is neat, and its visual representation of the song and the places where it can — but doesn’t have to — jump forward or back can be captivating. My Dearly Beloved has been particularly delighted with the results on “I Am A Camera”, by the Buggles, as it has many good edit points and can sound quite natural after the jumps if you aren’t paying close attention to the lyrics. I recommend playing that at least a bit so you get some sense of how it works, although listening to an infinitely long rendition of the Buggles, or any other band, is asking for a lot.
One question that comes naturally to mind, at least to my mind, is: given there are these various points where the song can skip ahead or skip back, how long should we expect such an “infinite” rendition of a song to take? What’s the average, that is the expected value, of the song’s playing? I wouldn’t dare jump into analyzing “I Am A Camera”, not without working on some easier problems to figure out how it should be done, but let’s look.
I have to imagine I’ve run across this before, but here’s a nice little page which allows one to search the (base ten) digits of π for any sequence of up to 120 digits that one wants. It searches the first 200 million digits of pi, which is enough digits that you can be reasonably sure that any string of six or seven digits you look for are there, and it’s not ridiculously unlikely that a string of ten digits in a row will turn up. The natural question is, why is this interesting?
People who’ve learned a bit about pi may have heard that it’s probably a “normal number”, that is, a number whose digits contain every possible finite string of digits within it somewhere. That suggests that finding any particular string of digits in pi is no more surprising than finding any particular word in a complete dictionary (if we imagine there’s a dictionary that ever did include all the words of a language). The story’s a little more complicated than that.
Since Scott Adams’s Dilbert hasn’t done anything to deserve my scrutiny let me carry on my quest to identify all the comic strips that mention some mathematical thing. I’m leaving a couple out; for example, today (the 11th) Rob Harrell’s Adam @ Home and Bill Amend’s FoxTrot mentioned the alignment of digits in the date’s representation in numerals, but that seems too marginal, and yet here I am talking about it. I can’t be bothered coming up with rules I can follow for my own amusement here, can I?
The Math Less Traveled has followed up the factorization diagrams post — rendering visually the integers multiplied together to get an integer — of a month ago with an expansion of the idea. This version includes not just arranging points into regular polygons, and polygons of polygons, but into colored polygons.
At least some of these charts, or charts inspired by them, belong on classroom walls.
My post on factorization diagrams from a month ago turned out to be (unexpectedly) quite popular! I got ten times as many hits as usual the day it was published, and since then quite a few other people have created their own variations. The purpose of this post is twofold: first, to round up links to a bunch of the variations, and second, to show off some new and improved factorization diagrams!
I should mention a couple of my inspirations: first, the book You Can Count on Monsters does something similar, though it takes a more artistic and representational approach, in contrast to my strictly geometric approach. My other inspiration was Sondra Eklund’s super-cool prime factorization sweater (and other prime factorization visualizations) which uses a different color to represent each prime.
Not long after I published my original factorization diagrams post, the Internet got right to work creating the…
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I dreamed recently that I opened the Sunday comics to find Scott Adams’s Dilbert strip turned into a somewhat lengthy, weird illustrated diatribe about how all numbers smaller than infinity were essentially the same, with the exception of the privileged number 17, which was the number of kinds of finite groups sharing some interesting property. Before I carry on I should point out that I have no reason to think that Scott Adams has any particularly crankish mathematical views, and no reason to think that he thinks much about infinity, finite groups, or the number 17. Imagining he has some fixation on them is wholly the creation of my unconscious or semiconscious mind, whatever parts of mind and body create dreams. But there are some points I can talk about from that start.
I apologize for being slow writing the conclusion of the explanation for why my Dearly Beloved and I would expect one more ride following our plan to keep re-riding Disaster Transport as long as a fairly flipped coin came up tails. It’s been a busy week, and actually, I’d got stuck trying to think of a way to explain the sum I needed to take using only formulas that a normal person might find, or believe. I think I have it.
As before, this is going to be the comics other than those run through King Features Syndicate, since I haven’t found a solution I like for presenting their mathematics-themed comic strips for discussion. But there haven’t been many this month that I’ve seen either, so I can stick with gocomics.com strips for today at least. (I’m also a little irked that Comics Kingdom’s archives are being shut down — it’s their right, of course, but I don’t like having so many dead links in my old articles.) But on with the strips I have got.
So, it’s established that my little series, representing the number of rides we could expect to get if we based re-riding on a fair coin flip, is convergent. So trying to figure out the sum will get a meaningful answer. The question is, how do we calculate it?
My first impulse is to see if someone else solved the problem first, for exactly the reasons you might guess. This is a case where mathematics textbooks can have an advantage over the web, really, since an introduction to calculus book is almost certain to have page after page of Common Series Sums. Figuring out the right combination of keywords to search the web for it can be an act of elaborate guesswork. Mercifully, Wikipedia has a List of Mathematical Series which covers my problem exactly. Almost.
Returning to the Disaster Transport ride problem: by flipping a coin after each ride of the roller coaster we’d decide whether to go around again. How many more times could I expect to ride? Using the letter k to represent the number of rides, and p(k) to represent the probability of getting that many rides, it’s a straightforward use of the formula for expectation value — the sum of all the possible outcomes times the probability of that particular outcome — to find the expected number of rides.
Where this gets to be a bit of a bother is that there are, properly speaking, infinitely many possible outcomes. There’s no reason, in theory, that a coin couldn’t come up tails every single time, and only the impatience of the Cedar Point management which would keep us from riding a million times, a billion times, an infinite number of times. Common sense tells us this can’t happen; the chance of getting a billion tails in a row is just impossibly tiny, but, how do we know all these outcomes that are incredibly unlikely don’t add up to something moderately likely? It happens in integral calculus all the time that a huge enough pile of tiny things adds up to a moderate thing, so why not here?
I suppose it’s been long enough to resume the review of math-themed comic strips. I admit there are weeks I don’t have much chance to write regular articles and then I feel embarrassed that I post only comic strips links, but I do enjoy the comics and the comic strip reviews. This one gets slightly truncated because King Features Syndicate has indeed locked down their Comics Kingdom archives of its strips, making it blasted inconvenient to read and nearly impossible to link to them in any practical, archivable way. They do offer a service, DailyInk.com, with their comic strips, but I can hardly expect every reader of mine to pay up over there just for the odd day when Mandrake the Magician mentions something I can build a math problem from. Until I work out an acceptable-to-me resolution, then, I’ll be dropping to gocomics.com and a few oddball strips that the Houston Chronicle carries.
Given that we know the chance of getting any arbitrary number — let’s say k, because that’s a good arbitrary number — of rides in a row on Disaster Transport, using the scheme where we re-ride if the flipped coin comes up tails and stop if it comes up heads, the natural follow-up to me is: how many more rides can we expect? It’s more likely that we’d get one more ride than two, two more rides than three, three more rides than four; there’s a tiny chance we might get ten more rides; there’s a real if vanishingly tiny chance we’d get a million more rides, if Cedar Point didn’t throw us out of the park and tear the roller coaster down first.
The Math Less Traveled over here shows off a lovely way of visualizing the factoring of integers by putting them into patterns inspired by the regular polygons. Some numbers factor into wonderfully obvious patterns; some turn into muddles of dots because integers just work that way. They’re all attractive ways to look at numbers, though.
In an idle moment a while ago I wrote a program to generate "factorization diagrams". Here’s 700:
It’s easy to see (I hope), just by looking at the arrangement of dots, that there are $latex 7 \times 5 \times 5 \times 2 \times 2$ in total.
Here’s how I did it. First, a few imports: a function to do factorization of integers, and a library to draw pictures (yes, this is the library I wrote myself; I promise to write more about it soon!).
primeLayout function takes an integer
n (assumed to be a prime number) and some sort of picture, and symmetrically arranges
n copies of the picture.
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So our scheme for getting a last ride in on Disaster Transport without knowing in advance it was our last ride was to flip a coin after each ride, and then re-ride if the coin came up tails. (Maybe it was heads. It doesn’t matter, since we’re supposing the coin is equally likely to come up heads as tails.) The obvious question is, how many times could we expect to ride? Or put another way, how many times in a row could I expect a flipped coin to come up tails, before the first time that it came up heads? The probability tool used here is called the geometric distribution.
Cedar Point amusement park, in Sandusky, Ohio, built in the mid-1980s a bobsled-style roller coaster named Avalanche Run, because it was the mid-1980s and bobsled-style roller coasters seemed like a good idea. My home amusement park, Great Adventure, had something called the Sarajevo Bobsled opened in that time because back then Sarajevo was thought to be a pretty good city apart from that unpleasantness seventy years before. But Cedar Point’s bobsled roller coaster had a longer existence than Great Adventure’s, and around 1990, it was rebuilt to something newer and more exciting, with a building enclosing it and a whole backstory behind the ride.
I haven’t time to write a short piece today so let me go through a fresh batch of math-themed comic strips instead. There might be a change coming to these features soon, both in the strips I read and in how I present them, since Comics Kingdom, which provides the King Features Syndicate comic strips, has shown signs that they’re tightening up web access to their strips.
I can’t blame them for wanting to make sure people go through paths they control — and, pay for, at least in advertising clicks — but I can fault them for doing a rotten job of it. They’re just not very good web masters, and end up serving strips — you may have seen them if you’ve gone to the comics page of your local newspaper — that are tiny, which kills plot-heavy features like The Phantom or fine-print heavy features like Slylock Fox Sunday pages, and loaded with referrer-based and cookie-based nonsense that makes it too easy to fail to show a comic altogether or to screw up hopelessly loading up several web browser tabs with different comics in them.
For now that hasn’t happened, at least, but I’m warning that if it does, I might not necessarily read all the King Features strips — their advertising claims they have the best strips in the world, but then, they also run The Katzenjammer Kids which, believe it or not, still exists — and might not be able to comment on them. We’ll see. On to the strips for the middle of September, though:
One. OK. We know that.
Every person who ever suffered through that innocent-looking problem where you’re given the size of a record and data about how wide the groove is and asked how many are on the side of the record and then after a lot of confused algebra handed in an answer and discovered it was a trick question has that burned into their brain, and maybe still resents the teacher or book of math puzzles that presented them with the challenge only to have the disappointing answer revealed.
This may be a generational frustration. I think but don’t know that compact discs and DVDs actually have concentric rings so that the how-many-grooves equivalent would be a meaningful, non-trick question; to check would require I make the slightest effort so I’ll just trust that if I’m wrong someone will complain. In another thirty years the word problem may have disappeared from the inventory. But it irritated me, and my Dearly Beloved, and I’m sure irritated other people too. And, yes, we’ve all heard of those novelty records where there’s two or three grooves on a side and you don’t know until fairly well into the performance which version you’re listening to, but I’ve never actually held one in my hand, and neither have you. For the sake of this discussion we may ignore them.
But the question we plunge into answering before we’ve noticed the trick is more like this: If we drew a line from the hole in the center straight out, a radial line if I want to make this sound mathematical, then it crosses some number of grooves; how many? Or maybe like this: how many times does the groove go around the center of the record? And that’s interesting. And I want to describe how I’d work out the problem — in fact, how I did work it out a few nights ago — including a major false start and how that got me to a satisfactory answer.
My title is an exaggeration. In eighth grade Prealgebra I learned many things, but I confess that I didn’t learn well from that particular teacher that particular year. What I most clearly remember learning I picked up from a substitute who filled in a few weeks. It’s a method for factoring quadratic expressions into binomial expressions, and I must admit, it’s not very good. It’s cumbersome and totally useless once one knows the quadratic equation. But it’s fun to do, and I liked it a lot, and I’ve never seen it described as a way to factor quadratic expressions. So let me put it on the web and do what I can to preserve its legacy, and get hundreds of people telling me what it actually is and how everybody but the people I know went through a phase of using it.
It’s a method which looks at first like it’s going to be a magic square, but it’s not, and I’m at a loss what to call it. I don’t remember the substitute teacher’s name, so I can’t use that. I do remember the regular teacher’s name, but it wasn’t, as far as I know, part of his lesson plan, and it’d not be fair to him to let his legacy be defined by one student who just didn’t get him.
Richer Ramblings here presents a rather attractive Venn diagram showing the possible combinations of eleven distinct sets. It’s a neat picture and one of the things that people who insist mathematics can be artistic are thinking of when they say it.
Venn diagrams are fairly good ways to visualize data, particularly the ways in which things can be parts of one or more sets simultaneously (or maybe part of no set). I find them most useful, in teaching, in doing probability questions, because so many questions about how probable something is amount to “how many ways can a described outcome happen”, and a nice, clean diagram can show just which outcomes fit which description. (“Coin comes up heads and the first child is a girl; coin comes up heads and the second child is a girl; coin comes up tails and the die roll is a prime number”, etc).
For that, though, I find their use kind of limited: if there are too many things happening (coin, child’s gender, die being rolled, goat behind door number two) the problem becomes one students’ eyes glaze over rather than try solving and I lose the thread of the question too. Worse, if there are too many possibilities, the number of lumpy circles I need to draw becomes smaller than the number of lumpy circles I can draw.
This picture does pretty completely away with the lumpy circles and goes in for much more involved curves. Some of the details are kind of small, but, this covers — at least if it was done correctly and I admit not testing — all the different ways that something can belong or not belong to eleven distinct sets simultaneously.
Thinking about the number of different subsets and shades that are needed — go on, how many are needed to give every distinct combination its own color (which isn’t what’s done here)? — makes me appreciate how choroplethy isn’t my thing.
Venn diagrams are cool, and extremely varied.
“If you think Venn diagrams are just a bunch of interlocking circles, think again. Pushing this iconic branch of mathematics to its limits reveals just how varied – and beautiful – these diagrams can be. This gallery showcases some of the wilder possibilities, including the most recent breakthrough in Venn geometry – the first simple, symmetric diagram to encompass a whopping 11 sets.” (New Scientist)
The picture above is the said first simple, symmetric diagram to encompass 11 sets, and yes, it is beautiful. “One of the sets is outlined in white, and the colours correspond to the number of overlapping sets. The team called their creation Newroz, Kurdish for “the new day”. The name also sounds like “new rose” in English, reflecting the diagram’s flowery appearance.“
Amazing stuff. Onwards!
So here’s my homework problem: On the original WiiFit there were five activities for testing mental and physical agility, one of which I really disliked. Two of the five were chosen at random each day. On WiiFitPlus, there are two sets of five activities each, with one exercise drawn at random from the two disparate sets, each of which has a test I really dislike. Am I more likely under the WiiFit or under the WiiFitPlus routine to get a day with one of the tests I can’t stand? Here, my reasoning.
GCDXY here presents images from the Apollo 13 flight checklist. This is itself a re-representing of images that Gizmodo posted when Apollo 13 Commander James Lovell sold the checklist last year, but I’m just coming across this now. And it nicely combines the mathematics and the space history interests I so enjoy.
The particular calculations done here were shown in one of many, many, outstanding scenes in the movie Apollo 13. However, the movie presents the calculations as being done on slide rule, when the computations needed are mostly addition and subtraction. It is possible to use slide rules to do addition and subtraction, but that’s really the hard way to do it; slide rules are for multiplication, division, and raising numbers to powers.
But considerable calculation for Apollo (and Gemini, and Mercury) was done without electronic computers, and the movie would have missed out on presenting an important point if it didn’t have the scene. So the movie achieved that strange state of conveying something true about what happened by showing it in a way it all but certainly did not.
Two hours after a service module’s oxygen tank explosion on Apollo 13, Commander James Lovell did calculations that helped put the ship back on course so that they could return back to Earth. They needed to establish the right course to use the Moon’s gravity to get back home. Check out the article on Gizmodo from November 2011.
We got a WiiFit, and a Wii, for Christmas in 2008, and for me, at that time, it was just what I needed to lose an extraordinary amount of weight. As part of the daily weighing-in routine it offers a set of challenges to your mental and physical agility. This is a pair drawn from, in the original release, five exercises. One is the Balance Test, measuring whether you can shift a certain percentage of your weight to the left or right and hold it for three seconds; the balance board, used for each of these tests, measures how much of your weight is where, left or right, front or back of the board. One is the Steadiness Test, about how still you can stand for thirty seconds and is trickier than it looks. (Breathe slowly, is my advice.) One is the Single Leg balance Test, trying to keep your balance within a certain range of centered for thirty seconds (and the range narrows at ten, twenty, and twenty-five seconds in). One — the most fun — is the Agility Test, in which you swing your body forward and back, left and right to hit as many targets as possible. And the most agonizing of them is the Walking Test, which is simply to take twenty footfalls, left and right, and which reports back how incredibly far from balanced your walk is. The game almost shakes its head and sighs, at least, at how imbalanced I am.
Mathematicians and philosophers are fairly content to share credit for Rene Descartes, possibly because he was able to provide catchy, easy-to-popularize cornerstones for both fields.
Immanuel Kant, these days at least, is almost exclusively known as a philosopher, and that he was also a mathematician and astronomer is buried in the footnotes. If you stick to math and science popularizations you’ll probably pick up (as I did) that Kant was one of the co-founders of the nebular hypothesis, the basic idea behind our present understanding of how solar systems form, and maybe, if the book has room, that Kant had the insight that knowing gravitation falls off by an inverse-square rule implies that we live in a three-dimensional space.
Frank DeVita here writes some about Kant (and Wilhelm Leibniz)’s model of how we understand space and geometry. It’s not technical in the mathematics sense, although I do appreciate the background in Kant’s philosophy which my Dearly Beloved has given me. In the event I’d like to offer it as a way for mathematically-minded people to understand more of an important thinker they may not have realized was in their field.
Kant’s account of space in the Prolegomena serves as a cornerstone for his thought and comes about in a discussion of the transcendental principles of mathematics that precedes remarks on the possibility of natural science and metaphysics. Kant begins his inquiry concerning the possibility of ‘pure’ mathematics with an appeal to the nature of mathematical knowledge, asserting that it rests upon no empirical basis, and thus is a purely synthetic product of pure reason (§6). He also argues that mathematical knowledge (pure mathematics) has the unique feature of first exhibiting its concepts in a priori intuition which in turn makes judgments in mathematics ‘intuitive’ (§7.281). For Kant, intuition is prior to our sensibility and the activity of reason since the former does not grasp ‘things in themselves,’ but rather only the things that can be perceived by the senses. Thus, what we can perceive is based…
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I hope everyone’s been well. I was on honeymoon the last several weeks and I’ve finally got back to my home continent and new home so I’ll try to catch up on the mathematics-themed comics first and then plunge into new mathematics content. I’m splitting that up into at least two pieces since the comics assembled into a pretty big pile while I was out. And first, I want to offer the link to the July 2 Willy and Ethel, by Joe Martin, since even though I offered it last time I didn’t have a reasonably permanent URL for it.
An interesting parallel’s struck me between nonexistent things and the dead: you can say anything you want about them. At least in United States law it’s not possible to libel the dead, since they can’t be hurt by any loss of reputation. That parallel doesn’t lead me anywhere obviously interesting, but I’ll take it anyway. At least it lets me start this discussion without too closely recapitulating the previous essay. The important thing is that at least in a logic class, if I say, “all the coins in this purse are my property”, as Lewis Carroll suggested, I’m asserting something I say is true without claiming that there are any coins in there. Further, I could also just as easily said “all the coins in this purse are not my property” and made as true a statement, as long as there aren’t any coins there.
Because there weren’t many math-themed comic strips, that’s why I went so long without an update in my roster of comic strips that mention math subjects. After Mike Peters’s Mother Goose and Grimm put in the start of a binomial expression the comics pages — through King Features Syndicate and gocomics.com — decided to drop the whole subject pretty completely for the rest of May. It picked up a little in June.
I mean to return to the subject brought up Monday, about the properties of things that don’t exist, since as BunnyHugger noted I cheated in talking briefly about what properties they have or don’t have. But I wanted to bring up a nice syllogism whose analysis I’d alluded to a couple weeks back, and which it turns out I’d remembered wrong, in details but not in substance.