Doing serious research level mathematics (or any other science for that matter) is almost impossible outside academia. Doing serious work is hard and mentally draining. Mere mortals can't motivate themselves to do it unless the incentives align just right (i.e. being in academia). Also it takes all your time so forget about it if you have another job.

I know this from personal experience. After I joined industry I thought I'll do physics research in my own time. In fact, I can't even keep myself up to date about current research. Most journal articles require a lot of effort to go through, and I don't have that kind of energy after a work day.

In math we have lots of journals devoted to recreational and more elementary problems. There is a lot of low hanging fruit out there.

For instance, compute the fibbonacci numbers modulo the square of a prime and compare to modulo a prime. Are there numbers where the first zero appears at the same point? We don't know.

I'm not saying the problem is easy. I'm saying one can make some progress without much theory.

I disagree. Look at 'simple' problems like Fermat's last theorem.

It took mathematics much beyond what was known at the time to get anywhere close to a solution.

And much beyond what 99.9% of people will ever hear about in their lifetime.

> I'm saying one can make some progress without much theory.

Claimed without evidence.

Sure, you can make trivial progress by writing a computer program to search, and the problem is "elementary", but "elementary" doesn't mean "can be solved without studying lots of techniques and theories", it means "can be solved without calculus"

A good example of this is Marjorie Rice, an amateur mathematician, who discovered four new types of tessellating polygons.

[0] https://mathtourist.blogspot.com.au/2010/06/tiling-with-pent...

I know the feeling, but don't forget that 'human flourishing' part of your life:

... the achievement of human flourishing, a concept the ancient Greeks called eudaimonia, or a life composed of all the highest goods. Su talked of five basic human desires that are met through the pursuit of mathematics: play, beauty, truth, justice and love.

(From TFA.)

Math is different. Even complete outsiders of mortal cognitive gifts can now and again generate publishable results.( I tried to find the reference but my google-fu fails me now - I do have actual published papers in mind.)

In physics these laymen are generally called 'crackpots' for a good reason because the context one needs to understand to generate new results is so huge.

Mathematics is different - all one needs is pen, paper, a mind that's attuned to generating logically whole statements and a problem that just won't let go of ones mind. The results are likely in geometry or some other approachable field and not in some more obscure subject.

That definition is circular. Of course research-level mathematics is done in academia, but there is a much larger field of applied mathematics that is done in industry.

That stuff has very little to do with 90% of academia math. Also, if you specialized and are interested in a certain field of math, the barrier to switch to another field is enormous. On the other hand, 90% of the applied mathematics done in industry is much, much simpler, and very often a trivialized version of what you do in academia, so one could imagine switching to be possible, but what's the joy of doing that?

(I have a MS in Math, my field was algebraic geometry. I considered a career in math research, but the industry was paying too well for it to be a reasonable)

Precisely.

In fact, in academia, in some fields (and mathematics is certainly among them), in some sense only "the best" contribute to pushing the field forward by working on the cutting edge.

And I think the author's point was that focusing only on that is an overly narrow view, and one can engage in maths in a recreational, joyful way, without aspiring to an academic career in it.