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For example, whenever I search for a proof of the Pythagorean theorem, I get a drawing of a geometric proof, yet we use the Pythagorean theorem to algebraically compute distance between points in an plane. Likewise sine cosine and tangent have geometric definitions, yet we determine their values not by drawing right triangles and measuring but by plugging angles into a function.

I've read about how Cartesian Geometry combined Geometry and Algebra, but how can we be sure that the two are compatible?

In other words, how do we know that we can just take a theorem like the Pythagorean Theorem, proved Geometrically, in the realm of rulers of compasses, and apply it to Algebra, in the realm of numbers?

Geometry seems too empirical. Why is it that just because we draw something on paper and it roughly works out that we assume it's true?

Thanks!

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    $\begingroup$ I hope this is not closed. It is a great question and deserves a careful answer. $\endgroup$ – WillO Jan 12 '15 at 1:45
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The "algebra" you mention is not modern, abstract algebra, but algebra as used in secondary school. Geometry is very intuitive, so using it at the secondary school level helps students to understand. That intuition seems to be built into our brains, since babies recognize faces and the like at very young ages.

However, in many higher-level classes, geometry is a result of analysis. For example, the sine and cosine functions are defined by infinite power series. The formula for the distance between two points is a definition, not derived from the Pythagorean theorem. A right angle is defined from the dot product of two vectors. The Pythagorean theorem then results pretty easily.

High school algebra is pretty much simplified analysis. So it is no wonder geometry and algebra are compatible: they both come from analysis, in their foundations if not in their presentation in secondary school.

All this does not answer the questions: why is the geometry we get from analysis such a good match for our intuition? And why is our mathematics such a good match for our universe? In my opinion, these matches are due to the fact that God is a mathematician. "Mathematics is the language with which God has written the universe" (Galileo).

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Why can we use geometric proofs in algebra?

...but how can we be sure that the two are compatible

The reason is this: Euclidean geometry, formulated in full strength as in Hilbert's axioms including the completeness axiom, says that Euclidean geometry in two or three dimensions is exactly coordinate geometry over the field of real numbers. Algebra over the real numbers is, in a sense, interchangeable with this strong form of Euclidean geometry.

You could say that the basic operations in algebra (addition/subtraction/multiplication/divison) are encodings of geometric information. If you have a notion of "Euclidean length," then addition tells us how the lengths of two collinear line segments are related to the total length when placing them end to end. The area of a rectangle is based on multiplication of side-lengths. You can also treat area as an additive quantity (when the areas are disjoint, of course.)

Consider also that the Pythagorean theorem can be formulated purely in terms of real numbers, and purely in terms of geometry. They are statements of the same fact, just interpreted through slightly different languages.

Basically, the real numbers are uniquely suited to idealized measurement. Whether it be length or area, they are the "most complete" field suited for the job.

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  • $\begingroup$ Thanks for this answer. I have a rough idea of what your saying but what mathematics would I need to fully understand this answer? $\endgroup$ – Bowen Jin Jan 13 '15 at 1:26
  • $\begingroup$ @BowenJin I would recommend you check out Hartshorne's book Euclid and Beyond from the library. $\endgroup$ – rschwieb Jan 13 '15 at 3:57
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There is a mathematical definition of a vector space, a vector space with inner product, a metric space. But there is not a definition (at least not one I know of) of what a geometry is. Yet there are at least three geometries which I hear every now and then, Euclidean Geometry, Hyperbolic Geometry and Paraboic geometry.

For the first one we have $5$ axioms (Due to Euclid) which tell us everything we can use to obtain a proof on euclidean geometry. It is possible to obtain a proof of Pythagoras theorem using these axioms.

On the other hand we have the plane $\mathbb R^2$ with the known metric and the algebraic properties of $\mathbb R^2$ as a vector field. We can verify that the plane satisfies all of the axioms Euclid asks for, hence everything that can be proved using Euclid's axioms is also true in the plane. Although I'm not sure if every statement that can be expressed in "Euclid's language" that is true in the plane is necessarily provable from the $5$ axioms.

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    $\begingroup$ Euclid's five axioms are very much inadequate for proving the theorems Euclid states — he implicitly uses extra assumptions that are true in $\mathbb{R}^2$, but aren't provable from the five axioms. Here's a more detailed explanation: math.stackexchange.com/a/328102/41415 $\endgroup$ – Daniel Hast Jan 12 '15 at 2:46
  • $\begingroup$ Wow, That's really interesting. $\endgroup$ – Jorge Fernández Hidalgo Jan 12 '15 at 2:48
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Let me add to the previous answers.

Geometry seems too empirical. Why is it that just because we draw something on paper and it roughly works out that we assume it's true?

We do not. We use axioms to prove theorems in any mathematical theory. Euclid was, as far as I know, the first one to try to build geometry on axiomatic system. This was a huge step, but his work wasn't completely satisfying because there were certain assumptions he made which could not be deduced from the axioms he has given (for example, he uses a fact that two circles of equal radii would intersect if they were close enough to construct equilateral triangle, which seems obvious, but couldn't be deduced from Euclid's axioms). It was Hilbert who was fist to give complete set of axioms of geometry as intended by Euclid and other Greek mathematicians, and that was not before 20th century. With set of axioms, you use deduction to produce theorems, such as Pythagoras theorem. Pictures/sketches are only used as a help to fully understand what is going on, and what do you actually need to prove what you want. If you carry out construction on paper, it is not a proof of the statement, rather you use abstract theory to prove certain construction will always produce what you want.

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  • $\begingroup$ What I should've said is "Why can we assume that something which is proven in one set of axioms (Euclid) applies to another (Arithmetic)" $\endgroup$ – Bowen Jin Jan 13 '15 at 1:28

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