This is a nice article. For those who have not yet read it (it's short, read it!), a one-paragraph summary: the author starts with a list of random numbers. Visualizing it (plotting the numbers, with the list index on the x axis) suggests / leads to (for the author) curiosity about how often numbers repeat. Plotting that leads to the question of what the maximum frequency would be, as a size of the input list. This can lead to a hypothesis, which one can explore with larger runs. And then after some musings about this process, the post suddenly ends (leaving the rest to the reader), and gives the code that was used for plotting.
Thank you so much for pointing this out! Experimental mathematics feels like a missing puzzle piece in which it makes so much more sense.
Quotes are from the wiki article you linked.
> As expressed by Paul Halmos: "Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does."[3]
I wish there were books on how people would describe their complete process (not only their proof) on how they figured things out.
> Mathematicians have always practised experimental mathematics. Existing records of early mathematics, such as Babylonian mathematics, typically consist of lists of numerical examples illustrating algebraic identities. However, modern mathematics, beginning in the 17th century, developed a tradition of publishing results in a final, formal and abstract presentation. The numerical examples that may have led a mathematician to originally formulate a general theorem were not published, and were generally forgotten.
Why is this the case? It seems like it doesn't benefit us other than saving some paper.
> The following mathematicians and computer scientists have made significant contributions to the field of experimental mathematics:
Fabrice Bellard
Donald Knuth
Stephen Wolfram
(among others)
--> This is so awesome, it also sheds some light into how these people think.
This isn't to defend how it's done, but the tradition has been that the "laboratory technician" skills are learned on the job. This is true of lab tech work as well. I've taught a number of summer interns how to solder, but it's not written up in any research paper. Of course that makes it hard if one isn't preparing to do it as a job.
By the way: the post ends with a conjecture that the maximum frequency is likely to be log n + 1 (with log here denoting the logarithm to base 10), but the more precise result seems to be that the largest frequency is:
((ln n)/(ln ln n))(1 + o(1))
with high probability (i.e., probability 1 - o(1)), where ln denotes the natural logarithm (log to base e). Empirically, the maximum frequency for n=1000, 10000, 100000 often seems to be, respectively, 5, 6 or 7, and 7 or 8.
This problem has applications in studying hash tables etc., and can be found under terms like "maximum load" with balls in bins, and proving this doesn't seem to be very easy. As the post says “the solution is likely not as trivial as it first looks”. The analysis may be hard, but these days if faced with a problem like this in the real world (e.g. we have a hash table of size M that will receive N entries in it, and we're curious about the likely maximum load), we can likely just experiment to find out. Even when the numbers are too large to run simulations directly, an in-between solution is to get a tractable expression (a recurrence relation using dynamic programming or whatever) for the closed form, and write a program to compute it.
This article is essentially an encouragement and a reminder of our ability to do experimental mathematics (https://en.wikipedia.org/w/index.php?title=Experimental_math...): there's even a journal for it, and the Wikipedia article on it is worth reading (https://en.wikipedia.org/w/index.php?title=Experimental_Math...). See also (I guess I'm just reproducing the first page of search results here) this article (https://www.maa.org/external_archive/devlin/devlin_03_09.htm...), these two in the Notices of the AMS (https://www.ams.org/notices/200505/fea-borwein.pdf, http://www.ams.org/notices/199506/levy.pdf), this website (https://www.experimentalmath.info), this post by Wolfram (https://blog.stephenwolfram.com/2017/03/two-hours-of-experim...), and there's even book by V. I. Arnold (besides a couple by Borwein and Bailey, and others).
Especially in number theory and probability, simple explorations with a computer can suggest deep conjectures that are yet to be proved.