History of mathematics I

TL;DR -- While browsing on education and technology, we settled on the medical example and look at suggestions for High School work given an interest in Medical School. That leads to mathematics since the normal will be computational involvement brought to us by GenAI and its impact on modern culture. As such, there is lots to look for that is not available yet. That will change as mathematics is lifted to public view. 

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This also could have been titled "computing and mathematics" given events of the past two years plus just what happened within the first month of 2025. However, the computer naturally comes out of mathematics (slide rule?, remember?, others?). That topic will be handled as needed. 

One focus that we have is supporting scholarly pursuits. In that sense and given our mission, much work has gone into the study of history and genealogy related to the experiences of the U.S. from early New England until now. We have supported the work of others in this regard through various means, mostly research. A goal has been to provide even more support of a financial nature. 

Lately, with GenAI now part of daily life for many, we have been increasing the attention given to technology; also, we consider that the U.S. and its 250th coming up fit into the various themes of governance of technology which seems to have been sorely lacking. With the emphais on computing being stronger now, we thought to look at people and their involvement especially as it pertains to the future. 

Education is a necessary theme. One result of GenAI's emergence has been that confusion has come about as to get involved or not. Many students adopted the technology early though overall there is no clear consensus on how such use ought to play out in an equitable sense. Now, after two years have lapsed, we see academic interest coming to fore. 

So, education needs to stay abreast of issues as well as adapt to new methods. We thought that it might be of interest to consider the experience and consquences of education with respect to a well-known field of operation, training of a medical doctor. As we started to browse, we ran into an interesting group and want to use them as a topic for further research. 

•  The National Society of High School Scholars -- NSHSS -- "NSHSS is a distinguished academic honor society, committed to recognizing and serving the highest-achieving student scholars in more than 26,000 high schools across 170 countries. With access to valuable resources and a network of more than 2 million exceptional peers spanning high school to college and career, NSHSS scholars embody academic dedication and outstanding leadership in their schools, workplaces, and communities."

Our first encounter with their site was this post: You Want to Be a Doctor? Here is the Average Salary for Each Medical Specialty. So then, if we look at that discipline which is well-known, one would look to see how to meet the requirements for doing the work. We all know of the academic work after the undergraduate college experience: medical school (internship); residency; further study. What about pre-college? 

Turns out that there was a NSHSS post that goes into this as it is a very good example: How to Prepare for a Future in Medicine Early in High School. This post gives an example course load with 4-years of mathematics, science and a foreign language. It is a typical mix for a scholarly pursuit with the newer emphasis on AP courses which exposes the student to college-level material. One new emphasis is on getting into the ways of calculus. We touch upon that below. Also, there are many more resources for supporting this focus, including preparation for testing. 

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One founder of NSHSS was of the Nobel family (Claes Nobel, Commemorative) whose prizes we started to pay attention to this year. Mainly, that was due to the Physics and Chemistry Prizes honoring computational work. The former was recognized in its association with mathematical physics. The latter included work sponsored by Google plus associated lab work. 

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The past two years we have seen GenAI/LLM come into daily life with impacts that range from good to bad and with unknown ramifications. One thing clear is that underlying what we observe and talk about is mathematics in action. Hence, that topic must be brought to general attention including supporting material that will make specifics known at all levels as needed. This is a new task, essentially. 

For now, let's look at one aspect of mathematics that the NSHSS has on its list: calculus. Other courses include preparatory work as well as advanced material. But, we thought to point to a fuller picture than one gets where the algebraic view is associated with geometry which is on the Freshmen list. 

With computing and its advances? Expect that geometry will be a part of daily life. It actually is already in a native sense. Parts of and events in our 3D world are associated with Euclid's insights from over 2000 years ago. Before the latter part of last century, students studies analytic geometry and calculus, rather than calculus alone which avoids some of the niceties for the mathematician. 

The following link is to a chapter of the material for a class in the History of Mathematics at UC Irvine: 8 Analytic Geometry and Calculus. Neil Donaldson of Scotland (Edinburgh and Bath Universities) is a visiting lecturer whose material covers many of the topics that will come into the discussion. 

As we have mentioned, AI of late is not due to some critter emerging via heated circuits with their buckets of bits. Not at all. We are seeing, finally, years of effort at computing mathematics paying off with remarkable results. That some of these astonish only show how much we do not know but could and will know. 

Now, all of this material covers a range of fields in mathematics, all of it undergraduate in scope. Now, a first step would be to bring this into a College 101 format. Given the ability now for presentation via the web, one could then point back to the view of the undergraduate mathematics major and then to various graduate specialities. Yet, at the same time, material related to the application of mathemaics would be (or become) available. 

Now, the general public? Or the C-Suite? That is work to be done. Domain focus? OpenAI's success in playing around with mathematical games does not mean that human expertise and talent will become less than worthy. On the contrary, people ought to (and will) be the focus. We already have touched upon this at lenght but need to organize the material.  

Remarks: Modified: 02/14/2025

02/14/2025 -- Two days ago, OpenAI pushed out a view, based upon aesthetics - finally, with respect to their models: OpenAI Model Spec.