Mathematics

In the thread on Jonathan Bartlett and priorities for Math education, I raised two comments that I think it would be profitable to further reflect on. First, from 33 on how the US National Academy of Sciences tried to classify Mathematics as a “science”: https://services.math.duke.edu/undergraduate/Handbook96_97/node5.html The Nature of Mathematics (These paragraphs are reprinted with permission from Everybody Counts: A Report to the Nation on the Future of Mathematics Education. ©1989 by the National Academy of Sciences. Courtesy of the National Academy Press, Washington, D.C.) Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, Read More ›

Our contributor JB asked to have a conversation on what is important in Mathematics, especially Math education. I shared some thoughts. >>Why not, let’s just do that, follow it up and see where it goes? For instance, I think a key insight is to find a useful, powerful definition of what mathematics is. If we understand what it is we are exploring, it may give us a deeper, richer view on how we may understand and apply it. And for this, I have come to the view that an adaptation of a view I was taught by my very first uni prof is key: Math is [the study of] the logic of structure and quantity. That, is, there are two Read More ›

On August 7th, News started a discussion on time’s arrow (which ties to the second law of thermodynamics). I found an interesting comment by FF: FF, 4: >> It’s always frustrating to read articles on time’s arrow or time travel. In one camp, we have the Star Trek physics fanatics who believe in time travel in any direction. In the other camp, we have those who believe only in travel toward the future. But both camps are wrong. It is logically impossible for time to change at all, in any direction. We are always in the present, a continually changing present. This is easy to prove. Changing time is self-referential. Changing time (time travel) would require a velocity in time Read More ›

Dr Carol Wood of Wesleyan University (a student of Abraham Robinson who pioneered non-standard analysis 50+ years ago) has discussed the hyperreals in two Numberphile videos: First: Extended: Wenmackers may also be helpful: In effect, using Model Theory (thus a fair amount of protective hedging!) or other approaches, one may propose an “extension” of the Naturals and the Reals, often N* or R* — but we will use *N and *R as that is more conveniently “hyper-“. Such a new logic model world — the hyperreals — gives us a way to handle transfinites in a way that is intimately connected to the Reals (with Naturals as regular “mileposts”). As one effect, we here circumvent the question, are there infinitely Read More ›