Jacob, the traditional definition of mathematics as "The science of quantity" or "The science of discrete and of continuous magnitude" is today inadequate, in that modern mathematics --while clearly in some sense a single connected whole-- includes many branches which do not come under this head.

Rick, I do not recall reading a definition of mathematics, but certainly I wouldn't accept it as being a science. Science refers to knowledge of the natural world, and mathematics does not exist as part of it. I have repeatedly referred to math as the heart of physics, whose laws are those upon which everything exists, including the pleasure of dialoguing in the Internet on physics.

Mathematics is a discipline, as philosophy is. The latter does not deal with numbers, but with musings on things that exist yet cannot be studied by means of experiments, nor accepted on faith. Mathematics is the discipline dealing with numbers in a way that can be understood and reproduced by means of axioms and theorems of gradual complexity. Numbers do not exist, making them 'infinite.' Letters do not exist either, so that they can be invented at will and combined to make infinite number of words.
Colors exist, and their number is finite, as are their possible combinations.

Mathematics is probably limited in its undertakings, because it is not a game with numbers but an understanding of the ways the measurable can be measured and quantities can be manipulated. Once manipulation moves into the realm of technology, "natural mathematics" moves into "applied mathematics." As such, the development of those applications create Artificial (man made) Sciences, an outstanding example being Computer Science.
The more mathematics can be applied to a Science, the more exact it becomes. Mathematics includes imaginary conventional numbers --such as negative numbers-- which at first sight are inapplicable for measuring purposes. The square root of a negative number, like the 'mystic' one sq.r. -1, serve convoluted exercises in purely theoretical mathematics.

The discovery of digitally expressing numbers lead to the art of digitizing associated mathematically expressible realities, such as audible mixed tones and their intensities in exactly reproducible mathematical expressions. Still, there are exact quantities that are not amenable to digital expression, such as pi, an exact quantity that exists only in the analog realm. But obviously any square is equal in area to a given circle, except that both geometric figures cannot be drawn together, for lack of an analog measuring instrument.
But a cube having twice the area of another one can not expressed digitally either, yet it can be drawn: trace the diagonal, and use it to draw a cube. This diagonal is the square root of the sum resulting from doubling the square of one of the cube's sides. Therefore, the hypotenuse squared is exactly twice the length of that side. But such a double-volume square can not be drawn using a ruler, because its markings are digital, not analog, obviously.

Contemporary accounts of the nature of mathematics tend to characterize it rather by its method than by its subject matter.
According to a view which is widely held by mathematicians, it is characteristic of a mathematical discipline that it begins with a set of undefined elements, properties, functions, and relations, and a set of unproved propositions (called axioms or postulates) involving them; and that from these, all other propositions (called theorems) of the discipline are to be derived by the methods of formal logic.

I would feel more at ease saying that the qualifier 'formal' derives from Plato's FORMS, which are taken as being the essence, understood as the abstract, non applied, IDEA. Any conclusion derived from premises, as per Aristotle’s syllogistics, belongs to the realm of 'material' (as contrasted to 'formal') logic. This is a classic paradigm of Materialism opposed to Idealism, embodying the polar divergence of Aristotle's scientific thinking from his teacher's inchoate understanding of the world. Aristotle must have been thinking of Thales while pretending to listen to what he was starting to consider as Plato's ramblings...

On its face, as thus stated, this view would identify mathematics with applied logic.

"Identify" is a very problematic term. I do not see the slightest connection between the two... Applied logic is a specific deductive process based on specifically defined premises, which in turn are the result of previous inductive --usually empirically gathered-- data. Mathematics is a discipline whose trunk has branched as the ancient sequoia it --figuratively-- resembles.

It is usually added, however, that the undefined terms, which appear in the role of names of undefined elements, etc., are not really names of particulars at all, but variables, and that the theorems are to be regarded as proved for any values of these variables which render the postulates true.

Quite a convoluted phrase. Not attempting to make it understandable, I will limit myself to say that I really do not care... Still, using symbolic elements and Boolean logic, all 'variables' can be taken care of.

If then each theorem is replaced by the proposition embodying the implication from the conjunction of the postulates to the theorem in question, we have a reduction of mathematics to pure logic.

I am reminded of the simple solution to the Gordian knot... What is pure logic? Mathematics is a natural constituent of the universe, of The Beauty. Homo sapiens discovered it and invents applications. Logic is exclusively invented by H. sapiens as a means of reasoning correctly to reach correct conclusions, required understanding the world, immediate or remote.

ADDENDUM: In Science, 28 May 1999, there is an article on "THE COSMIC TRIANGLE: Revealing The State of The Universe."
It is purported to "...represent the past, present and future status of the universe."
The three sides are : A) How much matter there is? B) Is it accelerating? C) Is it flat?
I am updated about the answers given to these questions, so that I did not learn much; what impressed me was the use of "Cosmic TRIANGLE", i.e., Geometry and BEAUTY, and D-SP's 'Geometrical Principle,' altogether...