> The problem is that the continuous is easier to imagine and prove results about [...]
Why would that be true? A cyclic group of order n, or even all of Z, seems to me easier to think about than the real numbers which employ philosophically tricky notions such as uncountable infinity, limits, convergence, etc. There are a lot of weird and counterintuitive results even in elementary analysis because of the way the real numbers work.
Of course, there's also discrete objects that are hard to reason about.
I think this is written from the perspective of a continuous mathematician or physicist, and the examples of discrete maths they have have in mind are numerical simulations, which are certainly harder to reason about than their continuous counterparts.
I also suspect that this is a fairly orthodox attitude among mathematicians - in "The two cultures of mathematics" (https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf) Tim Gowers says with more authority than I could:
> the subjects that appeal to theory-builders are, at the moment, much more fashionable than the ones that appeal to problem-solvers.
This isn't precisely the discrete/continuous split, but it's mostly aligned and the article puts combinatorics is firmly in the latter category.
I'm not sure that the "two cultures" split aligns with continuous vs. discrete maths. There's theory-building in continuous mathematics (abstract spaces, e.g. topological, metric, Banach, ... spaces) and in discrete mathematics (theory of finite fields), whereas both areas also have computational / problem-solving aspects (proving specific inequalities in the continuous case, or proving theorems about particular kinds of graphs in the discrete case).
It is true because the discrete world has the potential for more fine structure.
As an example showing that it is true, read https://math.stackexchange.com/a/362840/6708 explaining how it is possible that the real numbers are complete (all first order statements about them are either provably true or false) while the integers are famously incomplete (no set of axioms can prove everything about them).
ALL of the philosophically tricky notions that you list are only tricky when you mix in discrete notions like "integers" in your construction. And given that discrete mathematics gives us things like the Halting problem and incompleteness WITHOUT continuity being involved, shows that it is discrete mathematics that is harder.
> As an example showing that it is true, read https://math.stackexchange.com/a/362840/6708 explaining how it is possible that the real numbers are complete (all first order statements about them are either provably true or false) while the integers are famously incomplete (no set of axioms can prove everything about them).
> ALL of the philosophically tricky notions that you list are only tricky when you mix in discrete notions like "integers" in your construction.
That may be so, but people generally don't use RCF exclusively. Real analysis always looks at the real numbers as an extension of the natural numbers, you can't go anywhere without sequences and limits. So it seems disingenuous to say that continuous mathematics is easier just because RCF is complete.
> And given that discrete mathematics gives us things like the Halting problem and incompleteness WITHOUT continuity being involved, shows that it is discrete mathematics that is harder.
While this is true, it's also true that a purely additive theory of the natural numbers is complete and several other theories of discrete structures are too (for example, while group theory itself is not complete, it's certainly completable, unlike Peano Arithmetic).
Also, the halting problem and incompleteness aren't where weirdness ends. There's a whole other range of weirdness that happens only once you add in uncountable infinities, such as Skolem's paradox, the Banach-Tarski paradox or the undecidability of the continuum problem.
Why would that be true? A cyclic group of order n, or even all of Z, seems to me easier to think about than the real numbers which employ philosophically tricky notions such as uncountable infinity, limits, convergence, etc. There are a lot of weird and counterintuitive results even in elementary analysis because of the way the real numbers work.
Of course, there's also discrete objects that are hard to reason about.