Comment by photochemsyn | Hacker News Reader

Comment by photochemsyn | Hacker News Reader

Interesting work, but I think it neglects one of the profound distinctions in mathematics that people coming from a programming/compsci background often stumble over: that between the discrete and the continuous. This looks like a good discussion:

> "The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two - for example in computer models of continuous systems such as fluid flow - is a central issue in the applicable mathematics of the last hundred years."

http://philsci-archive.pitt.edu/16561/1/Discrete%20and%20Con...

With respect to programming and mathematics, this line is worth thinking about: The problem is that the continuous is easier to imagine and prove results about, but the discrete is unavoidable when it comes to calculation.

You should be looking into coalgebras. God they are so obscure but so insanely useful.

https://www.youtube.com/watch?v=XqywV-wkKSE

'discrete mathematics : algebra :: continuous mathematics : coalgebra'.

It's like the missing half of math and programming. Power series? Coalgebra. Combinatorics? Coalgebra. Continuousness? Coalgebra. Bases? Coalgebra. Control flow? Coalgebra [1].

I have a discord https://discord.cofunctional.ai.

[1] https://www.cl.cam.ac.uk/~vb358/articles/mpc15.pdf

> 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.

The continuous requires the law of the excluded middle, which makes it harder to imagine as it's not easy to "construct" an image of x in our mind purely from a proof of "not not x". That's why non-continuous (constructive) maths is also known as intuitionistic maths. For anyone who's not familiar with why constructive mathematics is non-continuous, it's because without the law of the excluded middle, only the computable real numbers (and analogues) exist, not the whole real number line that's present in classical mathematics.

The Banach–Tarski paradox is an example of an unintuitive result of the LEM, a paradox that doesn't exist in constructive mathematics.

It's common to say that constructive mathematics is continuous, as it cannot prove the existence of a discontinuous function. One can certainly reason constructively about the various things traditionally considered continuous mathematics. I find your opposite perspective on it a bit odd.

I wouldn't say that it cannot prove the existence of a discontinuous function. Rather it can prove that a point of discontinuity makes it not a function.

Can't you define a discontinuous function over some other set besides the reals, such as the set of all computable numbers? For instance, a step function is discontinuous at a single point. You can also define a function that's discontinuous at every point, such as the indicator function of the rationals.

I don't have any experience with constructive set theory, so maybe I'm missing something.

The current state of the metatheory of Banach tarski is that it follows from the ZF + Ultrafilter, but not ZF + dependent choice, and has very little to do with the excluded middle.

I'm sorry, but classical mathematics is perfectly able to handle continuous functions. And in fact it does so more easily. Which is why most of the important theorems were proven classically first.

Sure, Errett Bishop came along with https://www.amazon.com/Foundations-Constructive-Analysis-Err... and fixed that. But it is harder. And sure, as soon as you start looking at error bounds in numerical analysis, the classical shortcuts start to take work. But it is simply wrong to assert that the continuous REQUIRES the law of the excluded middle.

In fact every mathematician has been through the classical treatments of continuity in courses on real analysis and topology. Very few can say much that is sensible about constructivism.

I was in a similar situation while in college, as a math major. I was asked to run a tutoring session for the "statistics for psychology majors." Explaining continuous distributions -- why they were needed and had their own formulas -- was simply prohibitive. Even the students who learned calculus could solve the exercises, but didn't grasp the basics of continuous math unless they were really paying attention.

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