Category Archives: expository

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’, … Continue reading

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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 , and a … Continue reading

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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 … Continue reading

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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 … Continue reading

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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 1 to denote … Continue reading

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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 for boundaries of things, the symbol to symbolize … Continue reading

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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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