Leonhard Euler (1707-1783) was surely one of the most prolific of great mathematicians. Among his contributions, we need to mention two, each of which comes down to us as a single letter: the letter e and the letter i . If we were literally to "pay"tribute, we might do so in increments of $2.72, rounding up a bit the value of irrational e. We'll get to that. First: biography. Euler was born in Basel, Switzerland, so his life and work might fittingly be considered a riposte to the old anti-Swiss jibe (originally from The Third Man ) that Switzerland has produced nothing for all its years of peace and democracy, nothing more than the humble cuckoo clock. Since Euler’s day and because of his work, i has stood for the simplest of the numbers that Descartes had called “imaginary.” This i refers to the square root of -1. We don’t need to bother ourselves further with the question “ what is the square root of -1?” It is i , by stipulation. Also since Eul...
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Yesterday we discussed the limit of the process of compounding interest. As we move through even shorter periods, from semi-annual to quarterly to monthly compounding and so forth, and as we approach an infinity of infinitesimal periods, i,e. continuous compounding, we find the number 2.71828 arises to stare us in the face as a limit. This was Jacob Bernoulli's result. So why is the number called Euler's number, and sometimes just e? Did Euler name it for himself? And, if so, isn't that an injustice? Shouldn't it be called b? Well, Bernoulli found 2.71828... but he neglected to name it. Gottfried Leibniz, the man recognized as THE founder of calculus in the German speaking world, seems to have called it "b" in letters he wrote in 1690 and 1691. Perhaps this was a recognition of Bernoulli's role. More important, though, Bernoulli neglected to explore the properties of this number outside of banking. It was Euler (pictured above) who defined it as the...
Existential Comics, a very philosophy-literate humor website that I'm pretty sure I've invoked here before, has a funny item up about Bertrand Russell and Gottlob Frege. You can read it and behold it yourself here: www.existentialcomic.com/613. The gist of it is that Frege and Russell are portrayed as boxers, engaged in a championship match to become THE defining philosopher of a logicist foundationalism about mathematics. During the actual "bout" on this subject, Frege famously said "arithmetic totters!" At any rate, in the comic version of it, Russell and Frege are portrayed as both knocked out and flat on their respective backs at the end. I gather the point is that logicism, as to the philosophy of mathematics, tottered. And after, as folks like the barber who shaved every man in town looked on from the audience, I imagine (the comic does not go this far) that Cantor and Godel strode into the ring, bent upon developing a humbler conception of both m...
As I have discussed here before (notably in connection with the Vienna Circle), I believe there are limits to the applicability of the law of the excluded middle. One example: one can say with some plausibility that as recently as 1990 the validity of Fermat's last theorem was outside the scope of the law. The answer was not only not known, there was no clear procedure for solving it. So ... it was neither true nor false until such a proof had been created (not discovered, if one is going to take this view: created). Likewise with the Riemann Hypothesis, still unsolved to this day, though it helpfully provides us with a striking image above. A gentlemen who regularly comments on math issues in Quora wrote recently, "There are several fields of mathematics which only make sense on the premise that the Riemann Hypothesis is true, and a handful of others which only make sense on the premise that it is false." My point though is that n...
a n + b n = c n , where integer n ≥ 3, has no non-trivial solutions. This and, if all goes well, the next seven posts in this blog constitute an unprecedentedly ambitious project for me. I'll be attempting a very granular reading of the book I discussed in a more abstract fashion earlier, THE MURDER OF PROFESSOR SCHLICK: THE RISE AND FALL OF THE VIENNA CIRCLE. Let us start with the "rise" of that subtitle. In 1922 Schlick was offered and accepted the chair for natural philosophy at the University of Vienna. Edmonds calls this simple hiring "a turning point in the history of twentieth-century philosophy." The core of the study-and-discussion group that he seems to have formed almost immediately after that appointment consisted of Schlick himself, Neurath, and Hahn. Otto Neurath and Schlick had background and personality differences as far apart as the poles, as Edmonds tells it. Schlick was a gentile, Neurath a Jew. Schlick was soft-spoken...
Suppose there is a bank that will give you a great [underlying] rate of interest on your deposit: let us say 100% per year. Yes, I know that is an outrageous hypothetical, but humor me. If you give this "Bank of Euler" $1 on Dec. 31, then on the following Dec. 31 you can withdraw $2. Great! But, even better, the bank offers a half-year compounding. You can give the bank $1. on Dec. 31. On the following July 1 the bank will calculate your interest so far, and thereafter your interest will compound onto that new figure. This means that on July 1 they gave you half of the annual amount due you. In this case: fifty cents. So as of July 1 you have $1.50 in the bank. The rest of the year you earn NOT the other 50 cents, but half of the annual rate on that new number. This is $0.75. On Dec. 31, then, you will have the original $1.00 plus the mid-year $0.50 plus the end of year $0.75 in the bank. This is: $2.25. Better. By compo...
The relationship is complicated. Let us take four points. First. both were path-breaking mathematicians. Descartes’ invention of analytic geometry made possible, a generation later, Newton’s (and Leibniz') development of calculus. Second, they had very different ideas about matter and how different chunks of matter interact. Descartes wanted to get rid of the idea of “occult” causes and effects, or what some call teleology. To speak very roughly, mechanistic explanations involve pushing and teleological ones involve pulling. For Descartes, everything that happens in the material world had to be explained by things pushing each other around. There could be no “action at a distance,” no pull, just as things didn't happen because inanimate objects (or non-human animals!) wanted them to happen. So the Cartesians of the following generation argued against Newton’s law of gravity — in their eyes Newton was trying to bring back those occult causes, pulling at each other across vast d...
Recently, the physicist Sabine Hossenfelder, who maintains a wonderful blog, posted her thoughts about "infinity," its necessity as a mathematicians' tool, on the one hand, but its unreality from the point of view of the science of physics, on the other. Her point is that a physicist only regards something as real if it is necessary in order to explain an observation. Infinity, she says, will never fill that bill. She also went a bit into the mathematics of infinity, and the fact that not all infinities are the same. The real numbers are a different sort of infinity from the integers. I used the co mment section under her blog post to ask a question that has long been on my mind. I remember when I was a kid, many decades ago, some of us nerdy types excitedly read "One, Two, Three, Infinity" by George Gamow. Gamow covered some of the same territory you just did although I'm guessing progress has been made in the last half century or so. If I recall corr...
An out-of-the-way discussion of an out-of-the-way topic, the golden ratio. As many of my readers likely know, the "golden ratio" is the number found by dividing a line into two parts such that the longer part divided by the smaller part is equal to the whole length divided by the longer part. The very definition suggests the nesting of golden ratios inside one another, an activity that produces neat aesthetic effects. The number itself is an irrational one, 1.6183... Here's a further discussion: https://www.livescience.com/37704-phi-golden-ratio.html But the more out-of-the-way discussion I had in mind, my real gift for you today dear reader, is: https://www.math.cuhk.edu.hk/~pschan/uged1533/markowsky.pdf Geek out there.
In the fifth century in India, astronomer Aryabhata contemplated the fact that the moon can hide the whole of the sun during a solar eclipse. The two have the same visible size from the earth, then, although it was clear to Aryabhata that the sun was a good deal larger. He discussed this in a dispassionate mathematical way, one which offended the religious sensibilities of the day. Vedic scripture suggests that the moon is farther from the earth than the sun is, which Aryabhata realized was plainly wrong. Indeed, another great scholar of the day, Brahmagupta, who wrote both on scripture and on astronomy, both criticized Aryabhata for his impiety, AND then proceeded to build on Aryanhata's work in his own astronomical writings. There the matter lay until the 11th century, when Alberuni, an Iranian/Islamic scholar of the golden age of such scholars, became fascinated with Indian culture and history. He made this comment: "We shall not argue with him [Brahmagup...
This illustration is making its way around the more philosophical corners of the interwebs. In case I've given you an illegible variant of it, here are the captions, in sequence: 1. Shop for a new tie 2. Make macaroni 3. Do cardio 4. Don't inflate your ontology 5. Don't do it. 6. Vacuum the rug. The fifth cell includes an illustration of a unicorn and the square root of -1 as thoughts in the mind of the character doing various quotidian things. The warning, then, is not to attribute reality either to imaginary beings or to imaginary numbers. Leaving unicorns aside, I think there is something odd about the warning as it applies to numbers. The value of any number system, including the numbers we call real and those we call imaginary, consists entirely of its utility. So, for a pragmatist as to math, the square root of -1 stands on the same "ontological" footing as does -1, or as does 1. But I do like the cute minimalism of the illustrations here. ...
There is a dispute among economists that seems at first a mere superficial matter of prose style and presentation, but that may have a greater significance. The dispute is this: should economists use the paraphernalia of the quantitative sciences? Should they use formulae with lots of Greek letters, and graphs with clearly defined curves? One body of thought – found amongst self-identified Austrian school economists especially though not uniquely – suggests that such paraphernalia are inherently misleading. P.T. Bauer has phrased it well . Econometrics, he said, has “contributed to the disregard or neglect of evident reality” because the use of highly quantitative methods leads to unwarranted concentration on the variables that fit most readily into a formal analysis, and leads to “the neglect of influences which, even when highly pertinent, are not amenable to such treatment.” Bauer, the distinguished-looking fellow above, wrote this in 1987. He...
In yesterday's entry , I discussed a recent proof devised by a professor at the University of New Hampshire that there are "bounded gaps" between prime numbers. Specifically, Yitang Zhang has established that there are infinitely many pairs of primes that differ by 70 million or less. There are two fascinating things about this tidbit about which I wish to comment today. First, who the heck is Yitang Zhang? Second, infinity and size. Who the heck? One might naively expect the burning questions of a recondite field to be settled by the elites of the relevant expertise. Andrew Wiles, the fellow who proved Fermat's Last Theorem correct (though in a way that can't "fit into a margin") was a Royal Society Research Professor at Oxford University, specializing in number theory. Before that, he had been a professor at Princeton University in the early 1980s and a Guggenheim Fellow at the Institut des Hautes Etudes Scientifiques in France in the late 1...
All right. Let's really put our geek hats on and talk about prime numbers. A little over a year ago I wrote about an enjoyable evening I spent watching the musical Fermat's Last Tango, a fictionalized (and lyrical) presentation of Andrew Wiles' successful effort to prove that Fermat was right about a certain famously generalized form of the Pythagorean theorem. For purposes of the musical, Wiles is renamed Daniel Keane . Anyway, one of the conjectures that has acquired a good deal of importance in elite math-geek circles since Wiles' success is something called the " bounded gap conjecture" concerning prime numbers. As a refresher, a prime number is any number higher than 1 that can be divided only by 1 and itself. There are lots of prime numbers amongst the lowest counting numbers, but they thin out as one gets into the higher ones. 1 is by stipulation not a prime. Two and 3 are both primes. The first integer above 1 that isn't a prime, the...