the distinction between ‘relations of ideas’For the novel of the same name, see Hume's Fork (novel).
In philosophy, Hume's fork is a distinction that is drawn by David Hume between two different areas of human study:

All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic ... [which are] discoverable by the mere operation of thought ... Matters of fact, which are the second object of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing.
- An Enquiry Concerning Human Understanding
Hume's fork is often stated in such a way that statements are divided up into two types:

Statements about ideas - these are analytic, necessary statements that are knowable a priori.
Statements about the world - these are synthetic, contingent, and knowable a posteriori.
In modern terminology, members of the first group are known as analytic propositions and members of the latter as synthetic propositions. This terminology comes from Kant (Introduction to Critique of Pure Reason, Section IV).

Into the first class fall statements such as "2 + 2 = 4", "all bachelors are unmarried", and truths of mathematics and logic. Into the second class fall statements like "the sun rises in the morning", "the Earth has precisely one moon", and "water freezes at 32 degrees Fahrenheit".

Hume wants to prove that certainty does not exist in science.

First, Hume notes that statements of the second type can never be entirely certain, due to the fallibility of our senses, the possibility of deception (see e.g. the modern brain in a vat theory) and other arguments made by philosophical skeptics. It is always logically possible that any given statement about the world is false. (Note that statements like "either the Earth has precisely one moon, or not"...