eka-mark

This is the last thing that will have been written on this particular whiteboard! I’m moving to a new place, and today I taught my last lesson at the old place. I was teaching someone about diagonalization, which is more than an exciting enough note to go out on.

The role of a dictionary is to define unknown words by means of known ones. However there are terms, like left or right, which cannot be explained in this way. In the absence of a formal definition, material objects must be used to illustrate these terms: for example, we may say that the human liver is on the right side. […] Here we consider some cases where information cannot be explained verbally.

If you don’t have time to study math for an hour every week, try studying math for two hours every week.

…[T]he goal of mathematics is to develop enhanced ways for humans to see and think about the world.

William Thurston, foreword to The Best Writing on Mathematics 2010

This is now the official eka-mark blog.

TIL Newton probably didn’t know much about harmonic oscillators! The earliest known complete treatment of this system (after a groundbreaking first attempt by Huygens c. 1673) seems to have been in a 1727 letter from Johann Bernoulli to Daniel Bernoulli. The earliest published treatment was by Euler, presented about a decade later. Furthermore, there’s every reason to think that Newton, despite his discovery of several power series representations for trigonometric functions, never incorporated these functions into the calculus as such; it is again Euler who seems to have been the first to differentiate and integrate sine, cosine, etc.

This is the best resource for studying math that I’ve found in a while! It’s 300+ pages of flawed/incorrect proofs on topics including logic, analysis, and linear algebra. Each flawed proof is followed by a classification of its errors, and a corrected version.

Here, we show that rolling of wooden balls by bumble bees, Bombus terrestris, fulfils behavioural criteria for animal play and is akin to play in other animals. We found that ball rolling (1) did not contribute to immediate survival strategies, (2) was intrinsically rewarding, (3) differed from functional behaviour in form, (4) was repeated but not stereotyped, and (5) was initiated under stress-free conditions.

Many structures in mathematics are incomplete in one or more ways. …

… A fourth type of incompleteness, which is slightly less well known than the above three [algebraic, metric, logical], is what I will call elementary incompleteness (and which model theorists call the failure of the countable saturation property). It applies to any structure that is describable by a first-order language, such as a field, a metric space, or a universe of sets. For instance, in the language of ordered real fields, the real line R is elementarily incomplete, because there exists a sequence of statements (such as the statements 0 < x < 1/n for natural numbers n=1,2,…) in this language which are potentially simultaneously satisfiable (in the sense that any finite number of these statements can be satisfied by some real number x) but are not actually simultaneously satisfiable in this theory.

…[I]f one starts with an arbitrary structure U, one can form an elementary completion *U of it, which is a significantly larger structure which contains U as a substructure…. Furthermore, *U is elementarily complete; any sequence of statements that are potentially simultaneously satisfiable in *U (in the sense that any finite number of statements in this collection are simultaneously satisfiable), will actually be simultaneously satisfiable. … If U is the standard universe of all the standard objects one considers in mathematics, then its elementary completion *U is known as the nonstandard universe, and is the setting for nonstandard analysis.