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April 20

When is Math considered a Science?

For the Reference Desk, Math is a separate RD from Science, but that's for convenience only. I've heard Math described as the Handmaid to the Sciences, but that of course separates it from them. So when in general is Math treated as part of them and when isn't it?Naraht (talk) 12:57, 20 April 2021 (UTC)[reply]

"Handmaid"! Mathematics is the queen of the sciences! The term science has several meanings. Originally it referred to any discipline or branch of learning pursued by scholars, and German Wissenschaft has entirely kept this original broad sense, but the English term came later to be applied more narrowly to the natural sciences, governed by the scientific method of developing and experimentally testing theories. The three-way split into humanities, mathematics and science follows the currently most common English senses of these terms. --Lambiam 15:42, 20 April 2021 (UTC)[reply]

From my point of view at least, ignoring colloquial usage and the various poetic metaphors (handmaid of the sciences, queen of the science, language of science), the fundamental difference between math and science is methodology. Specifically, science uses inductive reasoning to establish truth and math uses deductive reasoning to establish truth. Of course the two types of reasoning can't be completely separated from one another in practice, but that's the ideal at least. As an example, to establish the Pythagorean theorem scientifically, one would gather a collection of right triangles of various dimensions, measure the threes sides of all of them, and verify that the sums of the squares of the legs is equal to the square of the hypotenuse to within the accuracy of the measurement. To establish the Pythagorean theorem mathematically, one starts with basic facts about the plane, the definitions of the concepts involved, and reasons deductively using the rules of logic to establish the truth of the theorem. This is basically the program carried out in Book I of Euclid's Elements. The method used in science is called, appropriately enough, the Scientific method, though a the scientific method in it's modern form is a relatively recent development (17th century) and one certainly wouldn't claim that science didn't exist before then. The method used in mathematics is called the Axiomatic method, which can said to have begun with Euclid, but really didn't reach it's current form until the 19th century. (There is some question on whether Euclid was the actual inventor or not, so much of the knowledge of the ancient world has been lost.) But actual mathematical research does involve inductive reasoning; many famous theorems started out as conjectures, and a conjecture is really just the name mathematicians give to a proposed truth that has only be found correct by experiment rather than mathematical proof. Also, newly hatched mathematical theories tend to be somewhat inductive rather than deductive in nature, I'm thinking Chaos theory and theories of Emergence. On the other side, physics is usually divided into the experimental and the theoretical, with the theoretical side being inherently deductive in method. So the boundaries are a bit fuzzy, which is to be expected when you try to categorize human activity. But conceptually, at least, the difference is fairly well defined, all things considered. --RDBury (talk) 16:36, 20 April 2021 (UTC)[reply]

How would you classify these??

In math, 1.5 and 1.50 are the same; in science they are different.
In math, 13 times 13 equals 169. In science, 13 times 13 equals 170 and 13.0 times 13.0 equals 169.

Georgia guy (talk) 17:14, 20 April 2021 (UTC)[reply]

Those are not statements about math or science, but about writing (by mathematicians and by scientists). --184.147.181.129 (talk) 19:36, 20 April 2021 (UTC)[reply]

There are at least two questions here, roughly "what is science?" and "what is math?". There's a broad sense to the word "science" that includes all systematic human knowledge (that's the original sense), and if you take that view, then mathematics is clearly a science, without needing to inquire further into what mathematics itself is.

If you're intending a narrower reading of the word "science", then you have to ask which narrower reading, and it starts to become important how you view mathematics.

The 20th-century philosopher most associated with the attempt to demarcate science from non-science is Karl Popper; see falsificationism. Falsificationism has been applied to mathematics, in particular by Imre Lakatos.

Getting away from views specific to individual thinkers, broadly speaking, there is a split between mathematical realism, which views mathematical objects as things in the world, and facts about them as "discovered", and formalism, which (as a caricature) treats mathematical facts as "invented", deduced from somewhat arbitrary axioms. If you are a realist about mathematics, and particularly if you treat mathematical knowledge as being discoverable empirically (see quasi-empiricism in mathematics), then it makes sense to consider math as a science even in a fairly strict sense. --Trovatore (talk) 19:49, 20 April 2021 (UTC)[reply]