“I never realised how hard actual mathematical proof would be. I’m currently studying different types of proofs in my calculus topic (which is insanely hard). I see what you mean by saying A-level maths does not teach you actual proofs, because the proofs they provide at uni are completely on another level than A-level”. Continue reading
Book method and my method
For the step 2 of the proofs by induction that f(n) is divisible by some number m for all n, the textbook says you should always start by working out f(k+1)−f(k). Continue reading
Amir Alexander’s book describes the arguments about the idea of “infinitesimals” – quantities smaller than any real number, but still bigger than 0 – as that idea came into increasing use in maths in the 16th and 17th centuries, culminating in the development of calculus by Newton and Leibniz.
There were not just arguments, but furious conflicts. The Jesuits, a powerful factor in the Counter-Reformation, vehemently dismissed infinitesimals as nonsense. So did Thomas Hobbes, who was probably in private an atheist but in political philosophy an advocate of absolute monarchy.
Alexander identifies the advocacy of infinitesimals with more liberal, democratic, scientifically open-minded and inquiring trends.
I think he makes too much of his thesis. He ends by arguing (with qualifications) that the anathematising of the idea of infinitesimals in Italy, and its relative welcome in England, determined the economic and intellectual stagnation of Italy in the next centuries, and the rise of England as a scientific and industrial centre.
He doesn’t deal with the fact that, in the terms argued at the time, the Jesuits were right. Infinitesimals, as discussed by 16th and 17th mathematicians, were a contradictory and illogical concept. Or with the fact that mostly, for two centuries after Newton, England, despite its more developed agriculture and then industry, was something of a backwater for mathematics, well behind France and Germany, and not hugely ahead of the Italy of Ruffini, Ricci, Levi-Civita, Vitali, etc.
He is right that mathematics often moves forward by building provisionally on dodgy foundations, and hoping for later research to make those foundations sound.
In the 19th century Cauchy, Weierstrass and others put a firm foundation under calculus, essentially by replacing the concept of infinitesimal by the concept of limit.
In the 1960s Abraham Robinson turned the tables by introducing a logically sound concept of infinitesimals with his hyperreal numbers. There is a clear and easy-to-understand introduction to his ideas at http://mathforum.org/dr.math/faq/analysis_hyperreals.html and a video on how that fits in with the history at:
John Conway’s surreal numbers also include a logically sound concept of infinitesimals.
You have a scale with two pans, so for example you can measure 2 grams exactly by putting a one-gram weight in one pan and a three-gram weight in the other. With what four weights can you measure any weight up to 40 grams? With what five can you measure any weight up to 121 grams?
(Hint: start with a simplified version of the problem. With which two can you measure any weight up to 4 grams? With which three can you measure any weight up to 13 grams?)
Prize: 200 Vivos and a Freddo for a successful answer and a good attempt at an explanation. Answers to Mr Osborn or Mr Thomas by Assembly on Tuesday 3 November. Continue reading
If you have a group of four people, it’s possible to have no subgroup of three all of whom know each other, and simultaneously no subgroup of three none of whom know each other. If there are five people at a party, is it possible that the party includes no subgroup of three who all already know each other, and no subgroup of three all of whom didn’t know each other before the party? If there are six, is that possible?
The prize was won by Joan Onokhua, who produced the best partial solution. Continue reading
We have to do our first exam, to test progress, in the week ending 16 October 2015. It’ll be a 50-minute test (not 90 minutes). Continue reading
Sec θ is called sec θ because geometrically, in the diagram above, it is the “secant” line (“secant”, from a Latin word meaning cut, is a line from a point cutting through a circle). Continue reading