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Comment by pwdisswordfish | Hacker News Reader

Pedantic distinction that exists only in English.

The distinction is significant, despite their relationship they're from totally different universes. Science is empirical: truth is validated by observation of the real world. Math is logical: truth is validated purely by internal consistency.

The idea that the domain of science is restricted to falsifiable theories, and is therefore fundamentally empirical, dates only to the early part of the twentieth century. Since the Enlightenment, and also in the much older Western classical tradition, the term science (and its closest translations in Greek, Latin, German, French, and many other languages) has referred much more broadly than that restriction permits, to all fields of knowledge established in a rigorous manner, whether deductive or inductive.

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...

Yea well, that's just like, if you're a mathematical formalist, man.

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 ...

Interesting. I'd like to learn more about this. Any pointers to how other languages and cultures don't distinguish between math and science? Does it apply to other things like the difference between induction and deduction as well?

In Italy math is considered generally part of science. In fact here at the universities math is part of "Mathematical, Physical and Natural Sciences Faculty" (roughly translated).

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 ?

In Polish the word for science is "nauka" (noun from verb nauczyć = to finish learning/teaching), and it also covers math. People still obviously understand the differences and there are more specialized (almost never used [1]) terms like nauka matematyczna and nauka empiryczna (doświadczalna), we just don't see the reason to specifically say "science other than math" very often, so unless you're talking about philosophy of science you'll just say nauka and be done with it.

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?

There are no pointers because just about every culture distinguishes between empirical and logical truths. I'm not sure if he's just being cute, but there's a very clear philosophical difference between math and science. This is not a pedantic distinction.

(Someone from reddit that "loves science" might think so, though.)

> just about every culture distinguishes between empirical and logical truths

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).

Nope. Descartes, Leibniz and Spinoza were rationalists. Hume, Berkeley, Locke were empiricists. This distinction wasn't formalized until relatively recently, but the intuition is very old.

In antiquity, Pythagoras and Plato fall in the former camp; Serapion of Alexandria and Philinus of Cos fall in the latter.

I have no expertise in ancient philosophy, so can't comment on these philosophers.

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).

First off, Hume rejected Descartes' cogito argument. Second of all, in Sceptical Doubts concerning the Operations of the Understanding[1], Hume makes a very rudimentary distinction between "Matters of Fact" (what we'd now call logical truths) and "Relations of Ideas" (what we'd now call empirical truths). Again, these classifications were intuited long before Hume/Descartes, but they exemplify them pretty well.

[1] http://www.livingphilosophy.org.uk/philosophy/David_Hume/Sce...

You claim that "the now well-established distinction between a and b was intuited long ago", but in the discussion refer to various a'/b' distinctions that separate the whole into quite different parts than today's a/b distinction.

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 a funny one, because as has already been pointed out, science means more than one thing in English. There's "rigorous study" and there's "natural science". It's assumed that the latter is rigorous and our best philosophical model for this is Popper's falsifiability. And whereas pretty much everyone except string theorists now accepts this model, it's pretty recent. Many ancient Greeks, for instance, favoured thinking hard about a problem over experimental evidence, either before or after the fact.

This is correct. In another language we'd refer to the distinction with a different word

In German, "Wissenschaft" includes all kinds of academic knowledge production, regardless if mathematics, natural sciences ("Naturwissenschaften"), humanities ("Geisteswissenschaften"), or engineering ("Ingenieurwissenschaften"). So, when translating the remark to German, it would be difficult to come up with a "different word" for the distinction.

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.

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