If you roll out a lump of clay into a snake, then tie a string loosely around it, can you contort the ends of the snake, without actually pressing them together, so that you cannot get the string off?
You can stretch the clay into a Medusa’s head of snakes, and tangle them as you like, again without letting them touch. If you are allowed to rest the string on the surface of the clay, then you can get it off: you just slide it around and over what was an end of the original snake.
Mathematics “has no generally accepted definition,” according to Wikipedia today. Two references are given for the assertion. I suggest that what really has no generally accepted definition is the subject of mathematics: the object of study, what mathematics is about. Mathematics itself can be defined by its method. As Wikipedia currently says also,
it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.
I would put it more simply. Mathematics is the science whose findings are proved by deduction.
This is about the ordinal numbers, which (except for the finite ones) are less well known than the real numbers, although theoretically simpler.
The numbers of either kind compose a linear order: they can be arranged in a line, from less to greater. The orders have similarities and differences:
Of real numbers,
there is no greatest,
there is no least,
there is a countable dense set (namely the rational numbers),
every nonempty set with an upper bound has a least upper bound.
Of ordinal numbers,
there is no greatest,
every nonempty set has a least element,
those less than a given one compose a set,
One can conclude in particular that the ordinals as a whole do not compose a set; they are a proper class. This is the Burali-Forti Paradox.
Note added, March 10–11, 2021. The bulk of this post concerns race in the theory of history, particularly the theory attributed to Johann Gotfried Herder (1744–1803). Not having read Herder for myself, I rely on the accounts of
R. G. Collingwood in § 2, “Herder,” of Part III of The Idea of History (1946),
Michael Forster in “Johann Gottfried von Herder,” Stanford Encyclopedia of Philosophy (summer 2019).
Somebody like Herder may introduce race as an hypothesis to explain history, but ultimately the hypothesis fails, by denying us the freedom that is essential to history as such. Nonetheless, Forster defends Herder as having
an impartial concern for all human beings … Herder does also insist on respecting, preserving, and advancing national groupings. However, this is entirely unalarming,
because, for one thing, “The ‘nation’ in question is not racial but linguistic and cultural.”
Change Collingwood’s word “race” to “linguistic and cultural grouping” then. I think his conclusion remains sound: “Once Herder’s theory of race is accepted, there is no escaping the Nazi marriage laws.”
This is about G. H. Hardy and Sylvia Plath: Hardy quâ author of A Mathematician’s Apology (1940); Plath, The Bell Jar (1963).
I first read Plath only recently, after encountering The Bell Jar by chance in the Istanbul bookshop called Pandora. Continue reading
This is mostly reminiscences about high school. I also give some opinions about how mathematics ought to be learned. The post originally formed one piece with my last article, “Limits.”
I learned calculus, and the epsilon-delta definition of limit, in Washington D.C., in my last two years at St Albans School, in a course taught by a peculiar fellow named Donald J. Brown. The first of these two years was officially called Precalculus Honors, but some time in that year, we started in on calculus proper.
Polytropy 