In High School, I suffered and stumbled my through all the geometry proofs. (I like the looks-like theory best). As if that wasn’t bad enough, in trig, we had to prove all kinds of oddball trigonomtric identities.
On behalf of geometry, I admit the theorems are important, but the proofs have no value. For trig, I thought knowing these uncommon identities might have some value in solving some very tough integrals, but no dice.
In short, after years and years of higher mathematics, these proofs were nothing but heartache. So, why are we forced to suffer so? Is there any practical application to these proofs? For example, can you picture Mike Brady, Great Architect of our times, first setting out to prove that two sides of his triangular (truss) roof are congruent, etc., prior to drafting up a single sketch? NO! What boss would ever expect that??? Besides, allowing for tolerances, they won’t be 100% congruent, anyway!
Because learning to do proofs teaches you to think carefully. One of the major weaknesses in the current education system is that children aren’t taught how to reason through and solve problems. At least not my nephews and nieces.
I agree that doing proofs is an extremely satisfying task.
[sarcasm] Yeah, man, I know whatcha mean, all through grade school, they forced me to do art class, coloring things, painting things, making stuff with clay, man, that was nothing but heartache. So, why are we forced to suffer so? Is there any practical application to that art stuffs? For example, can you picture Mike Brady, Great Architect of our times, first drawing his designs with those large sized crayons, prior to drafting up his sketches? NO! What boss would ever expect that??? Besides, he can have his secretary do any art work he wants done on the computer. So who cares? [/sarcasm]
The purpose of those proofs is twofold: First, is the beauty of the art itself, of the development of a complex structure from simple axioms and rules of logic. Second, the purpose of teaching those proofs to kids is to teach them how to think logically.
If you think that’s not needed, you listen sometimes to when politicians try to convince you of their position using faulty logic. Or you read some ads, or hear arguments, that rely on you swallowing faulty logic. The purpose of those proofs, if you need purpose, is teaching you how to THINK.
No, the architect doesn’t need to start out proving the triangles are congruent, he can stand on the shoulders of those who already did. But how do you think the first architects got there? ((ASIDE: Agreed, it was mostly observation, the first architects preceded Euclid; but Euclid formalized and organized the thinking process.))
In Catholic school I actually did a geometry proof (chord AB=CD) on 2 pieces of tracing paper and showed it to the nun! I was more of a right brainer, so my artistic side screamed “Of course it’s equal, the creative eye never lies!!”
I knew that the chords had something to do with the angles inscribed(?) or something like that to determine congruency(?). Obviously, since I do not work in NASA, math was not a driving force in my life. The nuns in my school that taught math were not easily amused by this outlook, nor my artistic attempts to solve a mathematical proof.
As a side note (many years later, in a galaxy far, far away) I actually stumbled upon a use for geometry in real life! Yup, I learned that firing any kind of mortar (or any type of indirect fire artillery weapon) involved geometry. I guess those fat penguins in my school got the last laugh.
I have a friend who is a mathematician. She works on somthing called Group Theory. About tewnty people in the world understand what she is doing . She can’t explain what group theory is (nothing to do with we lay people might imagine). It has no purpose what so ever.
But what is the purpose of soap operas od Gucci shoes or IMAX theatres? Surely the fact that we (taxpayers/governments/philanthropists/etcs.) can afford this kind of luxury is a sign that whatever kind of socio political system we are running must be working pretty well.
Consdider tha alternatives -stop all research that doesn’t have a practical purpose? Have all academics carry out government approved research? Close all the universities?
Mathmatical proofs may be useless, but they keep mathmeticians off the streets.
I once lost my corkscrew and had to live on food and water for several days
(W.C. Fields)
Proofs are necessary because all mathematics are built on a few axioms, i.e., they are held to be true as a matter of faith. If one of these axioms could be proven to be untrue, the whole ediface would crumble.
Advanced mathematics is outrageously complex. It delves into questions such as how the universe is structured in ten spatial and one temporal dimentions on scales many orders of magnitude smaller than a proton.
This level of complexity rests not only on the axioms, but also on the intermediate mathematics derived from them. There can be no confidence in the results of higher mathematics if the intermediate stuff is no proven. Hence, the importance given to questions such as Fermat’s last theorem.
On the other hand, I could be full of it since I never completed Calculus.
The philosophers have only interpreted the world in various ways; the point, however, is to change it. (Karl Marx, 1845)
ROFL, I sure got a kick out of that one Say, are you quoting this from someone else, or did you make that up yourself? I’m thinking of posting that at my local math department but I’d like to cite the credential
Anyway, back to the topic at hand… I just wanted to point out that in addition to the educational values of performing math proofs, there are very real applications as well. In physics, for example, your everyday logical reasoning and common sense can only take you so far.
As you delve deeper and deeper into the field, a place where the world no longer “makes sense”, math is the only tool you can rely on to guide your through all the mess. Therefore, knowing that some poor mathematician had spent a good portion of his life rigorously establishing the validity of the math you rely on is somewhat comforting to say the least
20 dimensions of space time… let’s see you tackle that without math…
As for higher mathematics, yeah, some of it has no application…yet. Recently there seems to be more of a push for applied mathematics, while pure mathematics is not getting the attention it deserves. People forget that there’s a symbiotic relationship between math and science. Science has been a catalyst for creating new math, but pure math has also been a catalyst for creating new science.
As for group theory in particular, I’ve heard a story that, earlier in the century, some physicists were discussing the curriculum at their university. One remarked to the effect of, “You know, we should stop teaching group theory–it serves no purpose.”
Currently, group theory has MANY applications in modern physics. It would be a tragedy if research into pure areas of math was ever neglected.
Even sticking with geometry, math is very useful. The ancient Greeks knew that Earth was round and even measured it fairly closely. Thus geometry fulfilled its name. I have always been impressed with that feat.
Do you think you could measure the Earth without geometry? You would need a damn large tape measure.
Probably a lot more than that. Group theory has lots of applications.
It is all based on the study of addition modulus n.
Bullhockey. It’s got plenty of applications: cryptography, physics (super string theory uses it, subatomic particle physics has used it to predict the existence of particles later confirmed by later experiments), basically any science that can model itself on finite integer addition.
Group theory was invented in the early 1800s by a Swedish schoolteacher if I recall correctly. It can’t really be called a product of the social welfare state. The money, whether from public or private sources, that supports mathematical research is really miniscule.
That’s the tough part of pure - as opposed to applied - mathematics. It is impossible to anticipate when it will come in handy. A lot of mathematics was invented to serve science, like integral calculus for classical mechanics for example, hell, most of physics relies various branches of calculus. But who would have figured in the 1800s that group theory would be useful in subatomic physics?
Mathmatical proofs are so essential to life! THey teach analytical thinking and reasoning. I often frame issues from great debates in terms of mathematical equations.
When I am working on my various artworks and inventions, I frame the proportions in terms of geometry. In fact, I used a protractor and compass to measure the angles when I was building a staircase in my house.
Proofs are one of the essential elements in analysis of any issue.
Jinx: When my father, who is a builder, puts down a foundation, he puts posts at each corner and measures carefully to see that the sides of the house will be the same. He then stretches twine along the diagonals and measures those as a double check to make sure everything is as accurate as possible.
Why can he do this?
Given: A rectangle ABCD (visualize a rectangle with A denoting the vertex in the upper left corner and the other letters assigned to the vertices as you proceed clockwise from A). Prove the diagonals are equal.
Since ABCD is a rectangle, or Saccheri quadrilateral, by definition AB = CD and AD = BC.
Again, by definition the angle at vertex A = the angle at vertex C (both are 90 degrees).
Therefore, by the Side-Angle-Side Theorem (I am not going to state this as it would take me at least 20 minutes to write, and I am not sure I remember it fully), triangle ABD = triangle BCD.
Every time I see this thread topic I think it reads: “Math Profs: Any Practical Application?”
Thanks for repated laughs, Jinx.
“I should not take bribes and Minister Bal Bahadur KC should not do so either. But if clerks take a bribe of Rs 50-60 after a hard day’s work, it is not an issue.” ----Krishna Prasad Bhattarai, Current Prime Minister of Nepal
Uh, that would be “repeated”. Seems I have a propensity for dropping vowels.
“I should not take bribes and Minister Bal Bahadur KC should not do so either. But if clerks take a bribe of Rs 50-60 after a hard day’s work, it is not an issue.” ----Krishna Prasad Bhattarai, Current Prime Minister of Nepal
Hmm…the geometry I took must have been more abstract. I’ll buy the “teaching logic” argument, but I strongly note! Because it was all pure logic, there was no measuring, no compasses, no protractors, and very little use of numbers. It was pure application of theorems to prove that angles were congruent regardless of the size of angle or length of a leg.
Those of you that state that they believe that they have done a proof by using string, traced sketches, measured, etc…that isn’t a proof in the classical sense. Proofs are performed with setting up a truths table, stating a given, and going from there. Building theorem upon theorem towards some logical conclusion.
Measuring, tracing, etc. That’s concrete! That’s the kind of proof I can accept.
That’s practical application of geometry. That’s what I was driving at. The practical nature is useful. The abstract side is beyond me.
By the way, did anyone have to do trig identity proofs? Not fun at all! (No, these types of proofs do not involve knowing an angle or leg of a triangle…)
We study geometrical proofs for two reasons. One you may regard as impractical, but the other certainly is very practical. First, you do proofs in order to convince yourself that the theorems are correct. If the theorems aren’t correct, what good are they? And don’t think that you’d be willing just to take the math teacher’s word for it that they’re correct! Second, the skill in “doing proofs” is the fundamental practical skill in math. Solving a math problem is basically the act of proving that your answer is correct–it is discovering a new (albeit highly specialized) theorem. The same approach applies to all rational thought.
The problem of doing experiments with string rather than proof is the limitation of practical examples. For instance, I draw a triangle with string in my backyard and notice that the angles sum to 180 degrees, no matter how I draw the triange. Then I draw (with string) a triangle that starts at my house in Chicago, goes to center New York, then down to Miami, then back to Chicago; and lo! the sum of the angles is more than 180 degrees. Without proofs, I conclude that BIG triangles don’t sum to 180 degrees but small triangles do. With proofs, I know that triangles on a positively curved surface (like the curvature of the earth) sum to more than 180 degrees, and triangles on a flat surface (like my back yard, roughly) sum to 180 degrees.
“Common sense” is all well and good but can lead to wrong answers (like the famous Monty Hall three door problem).
The methods of proof of geometry teach logic, which is applied to other areas of math, which is applied in every science. If it weren’t for the proof, we wouldn’t know the difference between a “good estimate” and a precise answer.
On group theory: Generalized group theory is a well known and well used branch of mathematics. What Colin’s friend works on could be a very tiny problem in the broader field, understood by only a handful of people. However, I thought that group theory was considered a “closed” topic, that everything to be discovered in that field had been?
Say, are you quoting this from someone else, or did you make that up yourself? I’m thinking of posting that at my local math department but I’d like to cite the credential 
