Reading Bolzano. Part 1.

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’:

Screenshot from 2017-08-29 14-15-15

 

(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.)

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Mathematical Aphorisms. Part 6.

It seems underemphasized that the parsimony of ZFC in the way of ‘types’ (having only one sort called sets) has the following practical and notational virtue: you can get by with a single type of the quantifier \exists, and a single type of the quantifier \forall; multi-sorted formalisms need multiple sorts for each of \exists and \forall, respectively. Why? Because otherwise the semantics do not work. (You can find a suitable example for yourself.)

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Reading Bolzano. Part 0.

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.

 

 

 

 

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Higher Trees Today. Part 0.

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.

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Vignettes. Part 13.

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 to denote a universe:

Screenshot from 2017-08-14 14-55-10

(p. 15 in George Boole, The mathematical analysis of logic : being an essay towards a calculus of deductive reasoning, 1847, Macmillan, Cambridge)

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Vignettes. Part 12.

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 \partial for boundaries of things, the symbol \rightarrow 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.

Screenshot from 2017-08-14 14-11-15

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Mathematical Aphorisms. Part 5.

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.

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