Jamesian Philosophy Refreshed

This is a puzzle created by the mathematician Augustus De Morgan. He offered this as a "proof" that 2 = 1, which is (as he understood) absurd, and dependent on sophistical reasoning. Can you spot the flaw?   1. Let  x  = 1.  2. Clearly,then,   x 2  =  x.   3. So  x 2  - 1 =  x  - 1.  Now, divide both sides of THAT equation by x-1.  That 'should' turn it into the following 4. x + 1 = 1.  But since we started by saying that X equals 1, we have now concluded that 5. 2 = 1.  QED.  That plainly cannot be.  But where is the sophism?  The answer begins with remembering a critical point in number theory: don't divide anything by zero.  Ever.  Any chain of inference that involves the division of anything by zero at any point is invalid.   The first proposition above tells us that x = 1. Later we are instructed to divide both sides of a certain equation by x-1.  This...

  I won't be saying anything new here next week. This will probably be the last you'll hear from me on 'Refreshed' until August 30. So for my final pre-break post, allow me simply to say that I enjoyed the moment in the movie The Gambler (1974) when the late James Caan discusses Dostoyevsky with his students.  Caan is a 'literature' instructor. His course work in vaguely defined, so he can discuss whatever books the screenwriters think suitable for that point in the plot of his extracurricular life.  The plot of this movie is very loosely drawn from the Dostoyevsky novel of the same name (where the protagonist is a private tutor).  In the movie, Caan is discussing another work of Dostoyevsky's, Notes from the Underground , and especially the passage involving the truism that 2 + 2 = 4. The protagonist rails against conformity, and reason itself, and says that he will assert his right to count 2 and 2 together as 5. It is absurd. Dostoyevsky, knows it, Caan...

  Continuing with our thoughts from yesterday.... What are we saying when we say we know that 45 + 303 = 348? That depends on what we understand numbers, and arithmetic, to be.  I think the sensible course is to adopt an operational/pragmatic view of mathematical knowledge according to which it is a shorthand for facts concerning the way the world works. In G.E. Moore's spirit, I might hold up a hand, "here is one hand." Then I might hold up another, :"Here, also, is a hand." Finally, with both of them in my field of sight, I can say, "there are two hands here."  So we can take 1 + 1 = 2 as settled. But 45 + 303 might be tedious to prove in that way, with some object other than hands (marbles?). Further, there would be the threat of false refutations. I might lose track of the first marble before I collected the 348th. Then I'd count them all and declare that the real answer to the question must be 347! A naive empiricism about such matters becomes...

Some more thoughts lifted from my ongoing work on my next book. -----------------------  The term “pyramid scheme” is sometimes used loosely as a synonym for “Ponzi scheme,” and their meanings are related, but they are in principle distinct. A pyramid scheme is a business model in which payments are promised to one level of participants for the act in bringing in another level of participants. You, dear reader, have no doubt received the simplest form of a pyramid pitch as chain letters or their e-mailed equivalents. “Send me a dollar. Also, send out six copies of this email to friends of yours, You will get $6 as a return on your $1 investment.” Each of those six people will soon be in the same situation if you comply. It isn’t exactly a Ponzi scheme, although it shares the unsustainability thereof. What’s the difference exactly? In a pyramid scheme, the originator can make a clean get-away. Once you’ve found your own six suckers, you’ve made your profit and the ...

What are numbers?   Are some numbers more real than others? More rational? Just … better? No, of course not! You reply. It is absurd to make a moral or quasi-moral judgment about or classification of numbers. Numbers are just … numbers.   Numbers are simply sums or aggregates. Each is a very short way of saying something about a collection of things, events, or people, as in the perfectly comprehensible sentence, "a number of people were hurt in the accident." There is no need to make heavy water out of this. But then, why do categories of numbers have the strange and sometimes stigmatizing names they do? Some numbers are called “irrational,” in contrast to the “rational.” Others are called “imaginary,” in contrast to the “real.” What is irrational about the former or unreal about the latter? It is a tricky question. But it is worth unravelling this knot. It is worth so even if you took no pleasure in mathematics at school, or since, and even if you are co...