### Manin on Foundations

#### Posted by David Corfield

Out of the series of observations made by Yuri Manin in Truth as value and duty: lessons of mathematics most relevant to us here is:

For a working mathematician, when he/she is concerned at all, “foundations” is simply a general term for the historically variable set of rules and principles of organization of the body of mathematical knowledge, both existing and being created. From this viewpoint, the most influential foundational achievement in the 20th century was an ambitious project of the Bourbaki group, building all mathematics, including logic, around set-theoretical “structures” and making Cantor’s language of sets a common vernacular of algebraists, geometers, probabilists and all other practitioners of our trade. These days, this vernacular, with all its vocabulary and ingrained mental habits, is being slowly replaced by the languages of category theory and homotopy theory and their higher extensions. Respectively, the basic “left-brain” intuition of sets, composed of distinguishable elements, is giving way to a new, more “right brain” basic intuition dealing with space-like and continuous primary images, both deformable and deforming.

Has mathematics learned better to employ its corpus callosum?

Posted at May 28, 2008 6:29 PM UTC
## Re: Manin on Foundations

Is that an emoticon on page 5 of Manin’s paper? I don’t think I’ve ever before seen an emoticon in a philosophy paper, though it’s bound to become common eventually.