Recent searches
Search options
In the first millennium CE, mathematicians performed the then-complex calculations needed to compute the date of Easter. Of course, with our modern digital calendars, this task is now performed automatically by computers; and the older calendrical algorithms are now mostly of historical interest only.
In the Age of Sail, mathematicians were tasked to perform the intricate spherical trigonometry calculations needed to create accurate navigational tables. Again, with modern technology such as GPS, such tasks have been fully automated, although spherical trigonometry classes are still offered at naval academies, and ships still carry printed navigational tables in case of emergency instrument failures.
During the Second World War, mathematicians, human computers, and early mechanical computers were enlisted to solve a variety of problems for military applications such as ballistics, cryptanalysis, and operations research. With the advent of scientific computing, the computational aspect of these tasks has been almost completely delegated to modern electronic computers, although human mathematicians and programmers are still required to direct these machines. (1/3)
Today, it is increasingly commonplace for human mathematicians to also outsource symbolic tasks in such fields as linear algebra, differential equations, or group theory to modern computer algebra systems. We still place great emphasis in our math classes on getting students to perform these tasks manually, in order to build a robust mathematical intuition in these areas (and to allow them to still be able to solve problems when such systems are unavailable or unsuitable); but once they have enough expertise, they can profitably take advantage of these sophisticated tools, as they can use that expertise to perform a number of "sanity checks" to inspect and debug the output of such tools.
With the advances in large language models and formal proof assistants, it will soon become possible to also automate other tedious mathematical tasks, such as checking all the cases of a routine but combinatorially complex argument, searching for the best "standard" construction or counterexample for a given inequality, or performing a thorough literature review for a given problem. To be usable in research applications, though, enough formal verification will need to be in place that one does not have to perform extensive proofreading and testing of the automated output. (2/3)
As with previous advances in mathematics automation, students will still need to know how to perform these operations manually, in order to correctly interpret the outputs, to craft well-designed and useful prompts (and follow-up queries), and to able to function when the tools are not available. This is a non-trivial educational challenge, and will require some thoughtful pedagogical design choices when incorporating these tools into the classroom. But the payoff is significant: given that such tools can free up the significant fraction of the research time of a mathematician that is currently devoted to such routine calculations, a student trained in these tools, once they have matured, could find the process of mathematical research considerably more efficient and pleasant than it currently is today. (3/3)
@tao In short, it is (just) another tool. The tool won't replace the expert. The tool lives in the toolbox of the expert.
@tao As one of my profs once said in an introductory logic class (in the context of proof assistants), we should use calculators only when we have enough intuition to sense that the answer given by it might be wrong (if it is wrong).
@DanielAricatt @tao the same healthy distrust of automation inspires the advice never to trust a test that hasn't failed.
@tao I somewhat disagree. By performing a trigonometrical calculation, you learn nothing. But when checking all the cases or look for counterexamples, you often discover something about the problem itself. This will be lost if we outsource everything to the machines. I loathe that I use GPS wherever I drive, I explore fewer places that way; a fun part of traveling died, and the same might happen in math. Arguably, there is of course the chance that the time saved can be used for something even more fun.
@domotorp In addition to continuing to teach the manual form of these skills in classes, I expect there to continue to be more recreational venues for these pursuits. For instance, modern chess engines far outstrip human grandmasters in capability, but human chess competitions are still a relatively popular activity. Similarly, with the advent of GPS, orienteering competitions are still a thing, as well as newer variants of these events (such as geoguessing or geocaching games) that actually incorporate modern tools into their structure. It is perhaps too early to say, but I believe high school math olympiads will still attract bright competitors even if AI tools become capable enough to solve most of the problems routinely. And such tools may enable new types of competitions that are currently not feasible to run (e.g., AI-assisted "math scavenger hunts").
Returning to the example of navigation, it is true that in the post-GPS era there has been a significant decline in basic navigational skills, for instance the ability to approximately determine North from a sighting of the Sun, Moon, or constellations (combined with some estimate of the time). This for instance may have exacerbated the recent phenomenon of flat Earth theories being entertained by some segments of the public, though it is likely not the primary factor. On the other hand, GPS is undeniably convenient; and while we always have the option to disable such features on our devices and try to navigate by more old-fashioned means, there really is very little rational incentive to do so (outside of the recreational environments mentioned earlier).