The ancient Greeks thought of all mathematics as being a branch of geometry. Addition of 2 plus 5 was interpreted in terms of a line segment of length 2 being put up against a segment of length 5 and producing a longer segment of length 7. Multiplication was seen in terms of areas. Division was seen in terms of "measure", that is, whether you could use a smaller segment to measure a bigger one. For example, 2 can measure 10, because you can use five 2's to measure 10. Irrational numbers such as the diagonal of a square were seen as somewhat problematic, and they didn't even really know about expressing numbers in decimal notation. This geometric reasoning also meant negative numbers and sometimes zero were viewed with suspicion, which lasted well into the fifteenth century in Italy in the algebraic works of Cardano, Tartaglia, et al during their work in solving the cubic and quartic formulae.

All of this can be seen in Euclid's Elements, for which modern translations are readily available. The Pythagorean theorem is Book I, Proposition 47. Most modern translations of Euclid are faithful to the original Greek, except that they tend to only use Roman letters to label figures (modern geometric notation freely mixes Roman and Greek letters).

As you can see, the Pythagorean theorem for the Greeks is purely a theorem of geometry. The Greeks' interest was in formulating a geometric statement and furnishing a geometric proof. There is some abstraction in the figure and how it's labelled which affords some generality, which can be comparable to an algebraic abstraction such as a^2 + b^2 = c^2. The ideas are there, but the way it's presented is different, purely geometric.

You asked about "complex mathematical formulas/proofs". You may have heard of the Euclidean algorithm, which is often taught in computer science courses today and is one of the earliest algorithms that we have a full description of (earlier mathematicians like the Babylonians had algorithms such as the square root, but they described it by way of examples instead of describing the algorithm itself). The Euclidean algorithm is also in the Elements, Book VII, Proposition 1. Book VII is about number theory, i.e. ordinary counting numbers. Today we would mostly express these results without geometry, but you can see here that even this result is stated in somewhat geometric terms, by how to find the greatest segment that measures two others. Note that geometry is very important to modern number theory in other ways, but we rarely state division in terms of measurement anymore.

I think this thread does a pretty good job explaining your question https://www.reddit.com/r/AskHistorians/comments/4j67ia/what_was_mathphysics_before_calculus_finishing/?st=IWOYANBA&sh=ee31582c

Edit: Original poster /u/ManicMarine

This response is mostly an example, as I think a response necessitates. I cited the parts that aren't as best as I could (damnit Jim, I'm a mathematician, not an historian).

The Greeks did mathematics via geometry. For them the square of a quantity actually meant the area of a square with side lengths that number. For example, two units squared is four square units, and the distinction between units and square units was of great importance.

I will provide a proof of the Pythagorean Theorem (a^(2)+b^(2) = c^(2)) that does not use symbols (and offset it as a quote to distinguish it from the rest of the discussion). Diophantus was one of the first mathematicians to give letters to unknown quantities (T.L. Heath, Diophantus of Alexandria: A Study in the History of Greek Algebra). They would have previously been referred to simply as 'first quantity', 'second quantity', etc. (D.E. Smith History of Mathematics, Vol. I and II). I will, in parentheses, enclose a clarification of the previous sentence in more modern language. That is not to say that they wouldn't have used symbols at all. They would name points and lines using letters (e.g. Euclid's Elements).

In order to do this, consider this image. The lengths of the sides of the large squares can be subdivided into a larger length and a smaller length (call the smaller length a, and the larger length b). Consider first the left image. The larger white square has area equal to the square of the length of the larger quantity and similarly, the smaller white square has area equal to the square of the smaller quantity (the big white square has area b^2 and the small white square has area a^(2)). The total area of two white squares is then the sum of these two quantities (a^2 + b^(2)).

Now consider the right image. In this image the right triangles have base lengths equal to the two quantities referenced above and the hypotenuse is a third quantity (call it c). Then the area of the white square in the right image is then the square of this third quantity. (that is, c^(2)).

Now, evidently the area of the two large squares (the left and right squares) is the square of the sum of the two original quantities (*(a+b)*^(2)) since each has side length equal to the sum of the two quantities. Furthermore, the area of each of the grey triangles (this number doesn't actually matter but let's call it X anyway) are equal as each of them have side lengths of the the first two quantities (a and b) .

Now, in each image the total area less the areas of each of the grey triangles is the white area (again, this quantity doesn't really matter, but is equal to (a+b)^(2)-4X). And having established that the total areas are equal and that the triangles each have equal areas, we must have that the white area in the first image is equal to the white area in the second image. (which we already established to be a^(2)+b^(2) and c^(2) respectively).

Now, recall that the third quantity was the length of the hypotenuse of a right triangle formed with side lengths equal to the original lengths. Thus we have shown that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.

You may notice that this takes considerably longer to do anything than if we have algebraic notation available to us. Again, I avoided using any letters in the above proof, but that is not what the Greeks would have done. To see an actual proof, you can click here for Heath's translation of Euclid's proof of the Pythagorean Theorem (along with a wonderful annotation).

I'm not a historian, but as a mathematician I think I can provide at least some clarification in a way that is hopefully not off-topic for this subreddit.

Whether algebra was created by the Arabs depends on what you call "algebra". What is certain is that the word "algebra" comes from the Arabic. There is now a fairly well-defined branch of mathematics called by that name, but one must be careful when reinterpreting in modern light what mathematicians of the past did in their own language. Contemporary mathematicians will tend to describe the Pythagorean theorem as "if a right triangle has sides a,b,c with c being the hypotenuse, then a²+b²=c²" or even "if x,y are two vectors in a Euclidean vector space such that x·y=0 then |x+y|²=|x|²+|y|²", but these formulations are obviously anachronistic in that they are statements of contemporary mathematics which are essentially equivalent, in our eyes, to the Pythagorean theorem.

In contrast, here is how the Pythagorean theorem is stated in Euclid's Elements (in an early 20th century translation, proposition I.47:

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

There are no variables in this statement, and it is not cast in what we would now call "algebraic" language: it says that the sum of the areas of the squares constructed on the two small sides of the right triangle is equal to that of the square constructed on the hypotenuse. The proof (here is a version online) is purely geometric: this square has the same area as this rectangle, that square as that rectangle, the sum of which equal the area of the third square. Euclid gives names to points (letters of the Greek alphabet, in sequence, which should maybe be thought of as numbers), rarely to lengths or areas; he never uses these names in the statements of the propositions, only in their proofs. The statements are always full sentences.

Euclid doesn't have a notion of "number" in the modern sense of the word, neither real numbers (even positive ones) nor natural numbers. Mostly there are lengths, areas and volumes, these being three different kinds of things; so there is no multiplication between them: the only way Euclid can speak of what we would now call a product is by speaking of the area of a rectangle whose sides are this and that length. In book V of the Elements (which is about proportions), there is a notion of "magnitude" (the Greek word is μέρος, which usually means a part a share, a portion) which is used to discuss common properties which can apply to all, like proportionality. In books VII and VIII (which are about arithmetic), there is a notion of "number" (the Greek word is ἀριθμός), which refers essentially to what we would now call an integer ≥2. So everything is formulated in these terms.

Perhaps more instructive than the Pythagorean theorem is the following proposition (V.25):

If four magnitudes be proportional, the greatest and the least are greater than the remaining two.

Nowdays we would state this: if x/y=u/v and x is the greatest and v is the least, then x+v>y+u. In modern terms, here is how Euclid's proof proceeds: since x/y=u/v, we also have (x−u)/(y−v)=x/y [this is a previous proposition, V.19] and since x>y we have x−u>y−v; but adding u+v we get x+v>y+u. In comparison to this, here is how Euclid writes it: to be able to subtract u from x, Euclid has a segment AB of length x and places a point G on AB at distance u from A, so that GB will be effectively x−u; and so on. Essentially everything has to be stated in this geometrical fashion.

Geometry, as I assume you know, is a branch of mathematics that studies shapes, sizes and the properties of space, and had been independently discovered by different cultures over time. The Greeks of Pythagoras's time had a solid understanding of geometry, although it wouldn't be for another few hundred years that Euclid formalized the science with his five postulates.

Using geometric techniques you can arrive at a number of mathematical truths, among them the Pythagorean Theorem. This essay looks at three different geometric proofs for the Pythagorean Theorem, including Pythagoras' own proof.

There's also this Vi Hart video, What was up with Pythagoras, where she:

1. Looks at the timeline of mathematical breakthroughs (2 minute mark).
2. Uses geometric techniques to prove that irrational numbers exist (3 minute mark).

She does a good job of explaining how interesting and complex mathematical ideas could be proved without the notion of variables or other mathematical tools we take for granted since the advent of algebra.

Geometric proofs were the standard method of computation until the invention of symbolic algebra. In fact, this is the reason that Isaac Newton refused to use anything but geometric proofs and constructions in his Principia. If you would like to see how the particular proof to which you refer was carried out, see Euclid's Elements, I. Prop. 47, where the Pythagorean theorem is stated (and later proved) as

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle (Heath translation).

Perhaps the more fundamental answer to your question is that these statements were proved using geometric constructions, which is the use of a compass and straightedge to prove theorems and propositions given a set of axioms (assumptions).

If you are interested and want to learn more, read the first few pages of the Elements and you will get a feel for this type of proof.

I would say that the example is not a very good one because "a²+b²=c²", by which I understood the Pythagorean theorem, isn't really algebra. Here a, b and c refer to specific values; it's really just a shortened way of saying "the sum of squares of the bases of a triangle is equal to the square of the hypotenuse."

On the other hand, it also isn't wholly true that the Greeks took no part in the invention of algebra. Diophantus of Alexandria, who lived in the 3rd century AD (some 700 years later than Pythagoras, in other words), first studied integer solutions to algebraic equations (such as a²+b²=c² itself, although the geometric content of this equation isn't really under consideration here). Although he never invented general methods for solving equations in either the integers or other systems, he was considered an important influence on Arab mathematics.