The big mathematics divide: between "exact" and "approximate" | Sociology and Pure Maths | NJW

Modern pure mathematics suffers from a major schism that largely goes unacknowledged: that many aspects of the subject are parading as "exact theories" when in fact they are really only "approximate theories". In this sense they can be viewed either as belonging more properly to applied mathematics, or as being essentially provisional; awaiting a more precise and logically viable treatment.

This crucial distinction actually cuts across many areas of modern pure mathematics. It starts of course with arithmetic, and the difference between counting and measurement, that is between intrinsically exact and approximate evaluations, but appears also in modern notions of algebra, topology, function theory, number theory and many other disciplines.

In this video we give an introduction to this important distinction, culminating in some unsettling thoughts about the logical validity of the "Riemann zeta function" and that most revered unsolved problem in pure mathematics: the Riemann Hypothesis.

This video is part of the Sociology and Pure Maths playlist found at https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua

The Playlist on Solving Polynomial Equations (yes we really do it!) is at Wild Egg mathematics courses YouTube channel: https://www.youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6

If you are interested in this topic, you might like to have a look at Curt Jaimungal's interesting interview with Prof Richard Borcherds, and my comment on that video re his response to a question I raised about the inexactness of "real number arithmetic". This video is at Curt's TOE channel: https://www.youtube.com/watch?v=U3pQWkE2KqM&t=1413s

Video Content
00:00 Exact versus approximate in mathematics
1:39 Associating applied maths to approximate values
4:58 Solving equations and ''real numbers''
12:05 Topological spaces
21:08 Functions
26:59 Number theory sigma and zeta functions
34:23 Riemann hypothesis issues

Видео The big mathematics divide: between "exact" and "approximate" | Sociology and Pure Maths | NJW канала Insights into Mathematics