The next installment brings few surprises. Bolzano argues for dividing knowledge into parts, and for interlinking those parts, within reason. It is a little surprising (to me) that Bolzano that early in his text defines what he means by ‘Wissenschaft’, and by a rather flexible definition: whatever is deemed worthy to be presented in a book he demands to be permitted to call ‘Wissenschaft’:
(This installment brought an innovation in form: the colors correspond to the portions of the text translated. Apart from that, the colors are not intended to mean anything.)
This serial will provide, in many and short installments, created every now and then, provide a new translation into English of, and commentary on, the German original of Bernard Bolzano’s 1837 classic book ‘Wissenschaftslehre’.
I omit the front-matter and table of contents, and start with the introduction.
Bolzano begins with a rather conventional (and, I think, even in Bolzano’s time, rather unsurprising) Ὁ βίος βραχύς, ἡ δὲ τέχνη μακρή:
To be continued.
For the time being, the installments of the serial ‘Higher Trees Today’ will focus on three kinds of higher trees:
- Contractible Spaces of Choices (c.s.c.)
- Diestel–Oum decompositions (d.o.d.)
- Kalai trees (k.t.)
- Masbaum-Vaintrob trees (m.v.t.)
Of course, while the latter three are quite unambiguously-defined and attested mathematical definitions, the first is a highly context-dependent notion (that anyone can apply to their favorite context, mathematical or not). Perhaps more precisely put: each of the latter three form a category, or perhaps even more precisely put, each of the latter three are structures in the usual sense of model theory, while the first is not known to either be a category or a class of structures, in any reasonable way. More coming soon.
There probably is no better illustration of the vagaries of notational fashion than the fact that nowadays, the numeral 1 is commonly used to denote the (conceptually very important) terminal category, while George Boole 170 years ago used 1 to denote a universe:
(p. 15 in George Boole, The mathematical analysis of logic : being an essay towards a calculus of deductive reasoning, 1847, Macmillan, Cambridge)
Did you know that mathematics will provide you with more-or-less standardized tools to usefully express commonly-occurring ideas? For exaple, the following illustration contains at least three examples of this: (0) the symbol for boundaries of things, the symbol to symbolize a process leading from a cause to an effect, and, of course, the good old Cartesian coordinate system, a tool usually credited to the publication of La Géométrie over 380 years ago, a tool of expressing ideas without which the following illustration would have taken much longer to construct.
Perhaps the best example1 of a sound and complete proof system for a general audience are the three Reidemeister moves.
1 Calling Reidemeister moves a ‘proof system’ goes back at least to Avi Wigderson’s 2006 ICM contribution.