The subject of geometry and algebra came up in this thread:
http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=185444
So this go me wondering, does geometry use algebra in its calculations? Or does geometry use calculations that are totally seperate from algebra?
Algebra is the basis for every other form of math that you’re likely to encounter (“you” being the average non-mathematician).
Geometry uses proofs to demonstrate relationships between lines, planes, and 3-D objects. If you think of how the subjects are generally taught in school: Algebra I -> Geometry -> Algebra II, you can see that Geometry is at least somewhat independent. This may have changed since I was in school (the middle ages:)), but I still think that tells you something. Algebra is all about solving equations. Two different things.
Somewhat of an oversimplification and I’m sure someone will nitpick my post, but I think that’s a good high level overview,
Geometry is mostly algebra with pictures. Like ultrafilter says, algebra is the basis for most all of math.
Nowadays, it’s common to mix the two as trigonometry: for instance, one might draw a geometrical diagram, and use algebraic relations for triangles - say, the cosine rule a^2 = b^2 + c^2 - 2bc*cos(A) - to derive an unknown angle or side.
However, classical geometry - for instance, Euclid - doesn’t use algebra. You draw the diagram and use imaginary ruler-and-compass constructions to deduce simple arithmetical relationships (at the level of “this line equals that line” or “this angle is twice that angle”) to derive the answer.
In 7th grade, I took Algebra 1. In 8th grade I took Algebra 2. Now, in 9th grade, I’m taking Geometry. Next year, I’m going to take Algebra 3-4 (presumably taught twice as fast as “Algebra 3” would be) so you’re pretty much correct. And yes, we are using algebraic equations to solve geometric problems and define lines and shapes.
Classical geometry does not use algebra.
However a model of geometry can be built from algebra. A point is denoted by a pair of real numbers. A line is denoted by the set of all (x,y) that satisfy ax + by = c, where a and b are not both zero. It can be shown that these points and lines satisfy the axioms of classical geometry.
What this means is profound. While it does not show that geometry is consistent, it does show that if algebra is consistent, then so is classical geometry.
Gödel has shown that we cannot prove that any (sufficiently complex) mathematical system is consistent. (Well, actually he showed that if we can prove a system is consistent, then it is not consistent.) So, the best we can achieve is to show that if we assume that algebra is consistent, then so is geometry. Something similar has been done to build algebra from set theory. Pretty much all mathematics can be built from set theory.
Well, actually, Euclidean geometry taken on its own is not sufficiently complex that GIT applies (I think), and there is a definition of consistency that can be proved within a system (Feferman gave it).
One difference is that geometry was the first math class I actually liked.
ultrafilter,
You may be right about geometry not being complex enough. Gödel’s proof requires that the system contain arithmetic. I will look into Feferman’s work. Thanks for the info.
Here’s a list of some of his papers. Might be a good place to start.
Tarski’s work is also worth checking out (for instance, here) - he concluded that Euclidean geometry is consistent and decidable.
Is Euclidean geometry the same as classical geometry?
Not quite. Euclidean refers to the body of geometry in Euclid’s Elements: 2D geometry using ruler-and-compass constructions only (i.e. constructions formed from intersections of straight lines and circles). Classical geometry is a bit broader in scope - the Greeks, for instance, used conic sections and various other curves to solve problems such as duplicating the cube - but still based on physical constructions rather than algebra.
Historically, geometry came way before algebra. For over two thousand years, people have been studying geometry either directly from Euclid’s Elements (the most famous and widely used textbook in the history of the world) or from other books based on his approach.
The credit for bringing algebra and geometry together usually goes to Rene Descartes. By representing pairs of numbers as points in the “Cartesian” plane, geometric figures like lines and curves can be described by algebraic equations; and algebraic techniques can be used to study geometry, or vice versa. (This is often referred to as “analytic geomtery” or “coordinate geometry.”)
I believe Godel’s caveat is that the system must include not only axioms, but also rules derived from those axioms. Axioms alone will not come into conflict (else, one of the axioms must be dismissed), it is these extended rules that conflict. The system is not only incomplete, because there are true statements which cannot be reached by the axioms and statements, but also inconsistent, because there are untrue (or at least, self-conflicting) statements that CAN be reached (such as, “This statement is false”. Therefore, any sufficiently advanced system is inconclusive.
Euclidean geometry incorporates five postulates:
-
A straight line segment can be drawn joining any two points.
-
Any straight line segment can be extended indefinately in a straight line.
-
Given any straight line segment, a circle can be drawn having the segment as radius and one end point as center.
-
All right angles are congruent.
-
If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
The fifth is spaced from the other four, because it’s negation forms non-euclidean geometry. The first four hold for all geometries.
Think of the axioms as the “brain” of the system because they control the system, and the rules as the “hands” of the system because they manipulate things for the system. The brain can sometimes set the hands in direct opposition to each other where there can, by definition, be no clear triumph.
Clasp your hands in front of your chest and push them against each other as hard as you can. Which hand will win? Whichever hand you let win.
Tim
Now, I thought that the fifth postulate was “Through a given point which is not on a given straight line, only 1 line can be drawn parallel to it”, and the non-Euclidean geometries were formed by changing 1 to 0 or 2.
That statement is equivalent to Euclid’s 5th postulate, and is usually given in treatments on the matter because it’s a bit more intuitive.
Actually, you could have a geometry in which the any of the first four fail to hold. I don’t know how interesting it would be, though.