Blum: “The proof is wrong. I shall elaborate precisely what the mistake is.”
arxiv.org
arxiv.org
For the collectors of useless knowledge: for a while, there was an attempt to establish the term "Strukturwissenschaften" (science of structures), covering mathematics, computer science, and system theory. But that term did not really stick.
edit: I was merely pointing out that the parent's comment was an opinion based the branch of mathematical philosophy that the commenter appears to subscribe. I thought I'd add some humor by wrapping it in a Big Lebowski reference. And now I've explained away any attempt at humor ...
It's in that sense that Gauss crowned mathematics the "Königin der Wissenschaften," the Queen of the Sciences. And indeed in German the term Wissenschaft in ordinary usage still includes mathematics, referring to all knowledge amenable to investigation (as opposed to received knoweldge) [0]. The terms for science in many other Western languages also retain this broader meaning to a greater extent that does contemporary English, and the same is true in many non-Western cultures as well.
The problem here is mainly a terminological one: in English we've found it useful to have a word that refers specifically to the Popperian notion of what counts as science (mainly, I think, to gain the power easily to distinguish "scientific" ideas from "non-scientific" bunk), but we haven't come up with a new term for the older, broader meaning of science as rigorously establishable knoweldge.
I should also point out that Popperian science involves plenty of deductive reasoning—we call it theory—so while it is true that Popper's notion of science requires empiricism, it is not correct to identify it with the distinction between empirical and deductive reasoning. At the same time, the practice of mathematics certainly does involve empirical reasoning, often resulting in a conjecture whose truth is subsequently subjected to rigorous deductive arguments.
(There is a section in the English Wikipedia article on mathematics that discusses this a bit more [1].)
0. https://en.wikipedia.org/wiki/Wissenschaft
1. https://en.wikipedia.org/wiki/Mathematics#Mathematics_as_sci...
(Someone from reddit that "loves science" might think so, though.)
Actually, having this distinction is quite young even in our own western culture: depending what exactly you mean, ~300 years (Kants distinction between "Synthetisches Urteil" vs. "Analytisches Urteil"), or 150 years(Gottlob Frege's 'invention' of formal logic). And modern philosophy of science is already doubting it again (e.g., W.V. Quine).
In antiquity, Pythagoras and Plato fall in the former camp; Serapion of Alexandria and Philinus of Cos fall in the latter.
Is 'cogito ergo sum' a logical or an empirical truth? For Descartes it was a logical truth (undoubtably true). However, from the point of view of Hume, it must be viewed as empirical truth, because it is based on an empirical observation. So if even philosophers of that time couldn't agree on the distinction between logical and empirical truth, I think I can safely conclude that this distinction was not at all an established part of the Western culture at that point of time.
Rationalists vs. Empiricists is what we call these groups of philosophers now, it is not that they grouped themselves in two schools of 'logical truth' vs. 'empirical truth'. In fact, Hume most probably believed that his own philosophical insights can be derived just by thinking, so they must be logically true. And still, this is quite different from what we today call logically true (i.e., formally provable).
[1] http://www.livingphilosophy.org.uk/philosophy/David_Hume/Sce...
I concede that expecting predecessors to split the whole exactly along today's a/b border would be asking too much. On the other hand, there is no objective measure that would allow us to test if a'/b' is similar enough to a/b to support your claim. So if you think it is similar enough, and I think it isn't, it seems to be more a matter of opinion than a matter of knowledge.
And my argument on "cogito ergo sum" was not about if Hume would consider it as true or false, but if he would consider it as logical or empirical statement. So your remark about that is true but irrelevant here.
It's like when you're talking about batteries in Tesla you don't specify they are reachargeable every time you use the word batteries.
I used this example, because the same difference in language works the other way for batteries. English has batteries (unsepcified) and rechargeable or non-rechargeable batteries. Polish has baterie (always non-rechargeable), and akumulatory (always rechargeable), and no general word for both, so you always specify which ones these are.
[1] Much more common distinction is "nauki ścisłe" (exact sciences - math, physics, etc) vs the rest.
As for induction and deduction I understand the terms and they exist in Polish (indukcja and dedukcja), but I don't see the point. Both are used in math and in sciences?
I am a bit surprised of the distinction; the term science comes from latin "scientia" which roughly means knowledge, so at least historically it was correct to consider math a science. Maybe it is a more recent (or just English) thing ?
Math is unique in that the standard of proof is more comprehensive and the domain is bounded but this is a good example of someone employing the scientific method and exhibiting good scientific community values.
In math you might also do that as a sanity check, but that's not the point. In math, the point is to actually _prove_ your theory to be correct, something which is absolutely impossible to do in science, where you can only prove theory to be _wrong_.
I get the impression that you're thinking of specific domains within science. Many theories are contextualized within something (e.g a theory of why there's lightning might be more domain specific than a theory of electromagnetism).
I don't disagree that things are proved within mathematical systems and scientific claims / hypotheses are categorically different (and weaker, I suppose).
But the author made a claim, discovered an error in their claim, and alerted the community to his mistake and signaled their intention to address it. That's not behavior that's exclusive to mathematics. Many scientists have made claims that they think the evidence supports, and then they realize that something wasn't correct, and they make the correction.
Cartographers / mariners used to think that California was an island: https://en.wikipedia.org/wiki/Island_of_California
They had solid evidence to think that's the case but then eventually they realized the mistake and they've corrected it.
Why do you say this? It is not supported by the link you provide.
This also means that mathematics lets us talk about universal properties ∀x∈S.P(x) (“for all x in S, P is true of x”), but science really only gives you existential properties ∃x∈S.P(x) (“there is some x in S for which P is true of x”). The scientific method is based on the idea that enough statements of the form ¬K∃x∈S.¬P(x) (“it is not known that there is an x in S for which P is not true of x”) resulting from a large number of convincing experiments can serve as evidence of ¬∃x∈S.¬P(x) (“there is no x in S for which P is not true of x”) which gives us an approximation of ∀x∈S.P(x).
E.g, "the moon is made of cheese". cf. Bertrand Russell's teapot.
As for applying that to math, yes that's true to some extent. And on rare occasion, mathematical proofs are found to have flaws in them years after they have become commonly accepted. I think there's an important distinction to be made, though, between the certainty of deductive reasoning, and the uncertainty of deductive reasoning as interpreted by humans.
If human-performed math were not a science, a retraction would never happen.
I don't understand this. You don't check a math proof against evidence, you check it for the correctness of the logical deductions from the axioms to the conclusions.
Here's a proof that there are infinitely many primes:
Take any finite collection of primes. Multiply them all together, and add one, giving a number X. Let P be the smallest prime that divides X (note that P might equal X). P cannot be in the original collection, so we have created a new prime. Therefore any finite collection of primes is incomplete, and so the collection of all primes must be infinite.
Suppose I want to check that for correctness. No amount of evidence will show that it's correct, so I really don't know what you are saying. Perhaps you could expand on your thoughts using this as a specific example.
The person I replied to said "this is how science works". I replied "math, not science", meaning in this case, it was an example of how math works (self-correcting through review). But that's also how science works.
I didn't mean to imply there were huge differences in the ontology of math and science, as practiced, although I found the subsequent discussion of the differences in their pursuits stimulating.
I see less difference in math and science than most, probably due to the influence of "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" iin which Wigner, one of the biggest players in quantum mechanics, points out that mathematical models/theories of physical systems often have unexpected extrapolative predictive ability, suggesting some relationship between math and physics (I believe some people would say that the universe has an underlying mathematical structure). https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...
For example, theoretical computer science is no science according to that distinction between math and science.
Nobody trusts the political scientists, sadly. There are too many bloody variables and collecting good enough data to build cumulative models is hard. We persevere, though.
I'm pretty sure those types of things, such as social sciences, can be tested and reproduced, but I'd imagine there are ethical concerns.
How do you reproduce Hitler's affect on politics and not end up on the receiving end of a tribunal in Hague?
In general day to day usage, science = natural/empirical science.
- The author has uploaded a [v2] with file size 0 bytes (see bottom of page); this constitutes retraction of the publication
- [v2] has a comment, shown approximately partway down the page, which is what this post's subject cites
You can click the [v1] link to get a PDF link to the now-retracted information. I really like arXiv's versioned publishing.
In other words, he might not have been building up the hype but instead was making sure he wouldn't have to retract a paper with a (more or less) obvious mistake in it. That's how science should be done rather than behind closed doors !
Many results in math establish a famous theorem by proving something more abstract that "easily" implies the famous result, like with Fermat's Last Theorem.
"Computer science is no more about computers than astronomy is about telescopes."
but that's disputed.
it was firmly implanted in people’s minds that computing science is about machines and their peripheral equipment. Quod non [Latin: "Which is not true"]. We now know that electronic technology has no more to contribute to computing than the physical equipment. We now know that programmable computer is no more and no less than an extremely handy device for realizing any conceivable mechanism without changing a single wire, and that the core challenge for computing science is hence a conceptual one, viz., what (abstract) mechanisms we can conceive without getting lost in the complexities of our own making.
But, most probably don't have a strong opinion on whether/which labels apply! They would say they just do research in TCS.
https://cstheory.stackexchange.com/questions/38803/is-norber...
I know there was a pool on how long till it was disproved. Who had 19days?
https://cstheory.stackexchange.com/questions/38803/is-norber...
E.g. If someone shared v1 before the author retracted, by visiting that link (even today) you would know nothing about what happened.