Hacker News Reader

Comment by brutusborn | Hacker News Reader

It’s because it’s the hardest field to bullshit. Everything is black and white and you can’t just make up fake data.

On the converse, psychology and economics seem to have the most problems in terms of reproducibility and fraud. This is because they are the easiest to falsify, and their results are interpreted according to less precise epistemology.

There’s no ‘Austrian school’ vs ‘Chicago school’ in maths (to my knowledge).

Also, the general public isn’t interested in pure maths, so there are less incentives to fake data so you can get a nice press release, or so you can publish your new book, or so that the government bureaucrat will subsidise your ‘research.’

View on HN

> On the converse, psychology and economics seem to have the most problems in terms of reproducibility and fraud.

If you think that psychology and economics have problems with reproducibility and fraud, you haven't seen the social sciences!

I somewhat disagree and would invite Georg Cantor to this discussion about acceptance towards new paradigms and black/white in Mathematics.

After all, Mathematics is not a natural science and hereby consensus must be reached.

What do you mean by the last sentence? My understanding was that most maths research used proofs that didn’t rely on any consensus?

I may well be wrong, would love some examples.

There can be uncertainty about unproven results (e.g., P vs NP), or about results that are too complicated for people to verify (there have been a number of these, where the proof essentially creates a new branch of mathematics).

I think they mean consensus about which axioms to accept. There could be competing schools of mathematics along those lines, I think there just happens not to be.

Hardly 'competing', more like (say) Commodore-64 demo Vs nVidia GPU demo scenes - you pick your arena and push what can be done with constraints (axioms).

In mathematics you might compare groups that work with Vs without the Axiom of Choice, you can can have one group satisfied by an existence proof that asserts X exists otherwise a contradiction must exist and another group that only accept existence proofs that demonstrate a means to construct an example of X.

The habeas corpus divide :-)

But is a Hilbert system easier to run in a computer since there are many axioms which clarifies the cases where contradictions or some inference rules are used which might be hard to program? For example, on the top of my head, in predicate logic, how would you encode an inference rule like exists elimination? It's tricky enough for me to think about, but to be able to encode a computer with it, is harder, and I can't remember if something like this falls under the undecidable problems category or not.

Ah that makes sense, but do they really matter? If I want to use calculus to design the thickness of a bridge, does the selection of fundamental axioms matter at all?

Don’t all roads lead to Rome? It’s like an engineer deciding to use SI or Imperial units, people have their opinions but at the end of the day both work.

In applied maths, probably not.

In theoretical maths, yes.

I may just be a biased engineer, but until you find a way to apply your theoretical maths, surely it doesn’t matter if there are theoretical differences between approaches?

In the same way some heterodox political philosophies may result in new moral systems, but it doesn’t really matter that these moral systems are different to the status quo, until someone uses it as an ideology for their revolution.

It’s probably good to have a variety of theory to choose from, except when your different theories result in different practical outcomes.

I think in theory, there could be very different axioms that both have so many open questions that competing schools vie for the best talent to explore 'their' space. Whether that's a situation likely to ever occur I have no idea.

If you did write a paper that assumed different axioms to the norm you would just state that you had done so! Because mathematics is axiomatic, mathematicians are happy to play around with axioms as long as it leads to something interesting.

I’ve heard it’s own version of pontoons though. In particular, that it’s becoming so dense that enough people aren’t revalidating the proofs. There’s been examples on HN of published proofs that existed for years and years before someone pointed out they’re wrong. To a non-mathematician this sounds like it’s own version of the replication problem.

This is somewhat overblown, and only really an issue for some particular theorems and obscure corners of mathematics. And this isn't a recent problem, Hilbert's proof of Nullenschtatz in the early 20th century had logical errors, but is otherwise true.

If a proof is used by 3 people it won't have much scrutiny, but once major results start to be based on the proof, it'll get reviewed more carefully and either accepted or rejected in the long run. The ABC conjecture is probably the biggest example.

> There’s no ‘Austrian school’ vs ‘Chicago school’ in maths (to my knowledge).

see Brouwer–Hilbert controversy ?

Neither Brouwer nor Hilbert claimed the other was wrong, only which axioms they preferred to use.

This is not true at all, Brouwer truly thought the original Aristotelian concept of the "law of excluded middle" was epistemologically unfounded. I would recommend his original paper introducing intuitionistic mathematics but it's very dense. You can refer to: https://plato.stanford.edu/entries/brouwer/

Later, he argued he found mathematical proofs of counterexamples of LEM. From above source:

> “Intuitionist Reflections on Formalism” of 1928 identifies and discusses four key differences between formalism and intuitionism, all having to do either with the role of PEM or with the relation between mathematics and language. Brouwer emphasises, as he had done in his dissertation, that formalism presupposes contentual mathematics at the metalevel. He also here presents his first strong counterexample, a refutation of PEM in the form ∀x∈R(Px∨¬Px), by showing that it is false that every real number is either rational or irrational. See the supplement on Strong Counterexamples.

Reply on news.ycombinator.com