Or mathematics is the idealization of the world.
I seriously can’t tell how the OP intends us to match up the categories (3-4-5, 3-4-5 is WRONG) with the analogized categories (evolution, creationism). I think the confusion comes from the fact that his analogy is presented from the position of a complete moron who doesn’t comprehend math at all. Is he arguing that we should “teach the controversy”? Or is he arguing that only a moron would want to do so?
I’m so confused…
What about the 5-12-13 triangle?
Splitter.
Maybe the OP is advocating that schools recognize answers that are calculated using floating point operations on an old Pentium processor from 1994. I imagine another issue up for debate is whether the current year is 2009 or 1909.
Well, I never trusted triangles anyway. So aggressive, all lines and pointy bits. And all that empty space in the middle. I just tried it out, I got my crayons out and drew a triangle, and guess what: The sides weren’t 3, 4 or 5 at all! And there were no right angles. In fact, one of the lines wasn’t even straight! The OP is right, it’s all a big lie!!!
There must be some way of protecting the children from all this brainwashing, maybe they could wear tin foil helmets in school or something.
Triangle Man hates Person Man.
To take the OP perhaps more seriously than needed: (:))
Obviously 3 squared (9) plus four squared (16) equals five squard (25) by any definition of arithmatic worth discussing. So more properly speaking, the question would be whether a triangle with those sides would really have an exactly 90-degree angle, not infinitesmally greater or lesser than 90 degrees. Actually, more broadly* the question is whether all right triangles meet the pattern of (side1)squared plus (side2)squared equal (side3)squared, whatever the actual side length.
The answer, proven long ago, is “yes”. Given some axiomatic presumptions about plane geometry and about what constitutes logical proof, it can be shown that by definition of the terms used than the theorem must be true.
To my mind, what’s even more interesting than 3²+4²=5² are the limitless series of triangles in which the two short sides differ by 1, coming ever closer to being almost equal in length. I spent quite a bit of time rediscovering how these calculated. The next in the series is 20²+21²=29², the one after that is 119²+120²=169², and so on. I once worked out a table of these to examples where two astronomically long sides would only differ by the width of a hydrogen atom.
*more broadly still, any triangle’s sides can be related by a formula which only reduces to exact squares for the special case where the angle opposite the longest side is 90 degrees. This forms the basis of trigonometry.
but what’s the difference between reality and mathematics?
“reality” is what i see and measure outside of myself? and mathematics is at the core of where who i am actually touches reality head on?
mathematical logic is in my being?
mathematical logic is what i know for sure, but “reality” is what i can only make approximations of?
if the point in which whatever i truly am touches reality is not itself part of reality, then what is it?
what do you make of the electrical light energy that makes up logic in a human’s head? is there any reality to it, or is it all complete fiction?
where does our idea of the number 1 come from? what is the explanation of our idea of the number 1?
what does the number 1 mean, on its own, without any units put to it? what does it mean that we can use the number 1 without ever assigning any units to it? if the number 1 does not have any real existence in the “real world” without units, where does the number 1 come from? does it not come from reality? is it not real? how can we talk about it if it’s not a real part of reality?
how do you explain the mapping of the 3 4 5 triangle on a real sheet of paper using real ink (even if it’s not exact, it’s close enough that it appears amazing, because it maps to something real in our heads)
are the light particles that are bouncing in our heads not real? what are they? and how do you explain them? how on earth would the real world… the sun and the earth, go about putting light particles in your head so that you can think of a 3 4 5 triangle?
why would the universe create a human through evolution and place the thought of a 3 4 5 triangle in its head?
isn’t your head a consequence of real life?
how do 3 4 5 triangles exist in real life in your head?
how do we know what a 3 4 5 triangle is if it doesn’t exist, at least in our head. and if it exists in our head, then can it be said that what is in our head is a real 3 4 5 triangle. we hold light particles in our head in the shape of a 3 4 5 triangle?
how can we form light particles in our head in the shape of an exact 3 4 5 triangle when we don’t even know the exact angles of the 2 non-90 sides?
how can a real 3 4 5 triangle exist in my head (i know what it is) and not exist in my head (i don’t know the exact angles) at the same time?
and if the real 3 4 5 triangle does not exist in my head, then shouldn’t it be said that there is no truth at all to a 3 4 5 triangle? and nobody should even mention them or use them, because they are a myth?
if what’s happening inside my head is not real life, then what is it? why would evolution go through millions of years of struggling just to create something that’s not real? why is the brain not real?
What about the 20-21-29 triangle? (On edit, Lumpy beat me to it.)
I think most of us are beginning to doubt the reality of yours, so you’ve got a point there.
I forgot I was dealing with the Sperm Guy :).
As to your first question (what is “1”), well, good question. We probably (IANA mathematical historian) got the idea for it by making notches sticks to keep track of animals. Adding up “1s” gives you every other number. But then, what is a number? It is also a mental construct, eminently useful to survival but still theoretical. You can show me nine of something, but you cannot show me plain ol’ nine without making up a symbol for it. But once we have the symbolism, we can do all sorts of things. In fact, one of the most important concepts of modern mathematics is that of notation. It’s visual shorthand that everyone agrees on to communicate concepts; if you want once instance demonstrating how useful notation is, look up Liebniz notation. It has made understanding derivatives easier for students for centuries!
On your second point, I thought I covered this in my last post, but here goes. We use the idea of 3-4-5 triangles (and perfect circles, squares, lines, etc.) because they are useful. If you build me something that conforms to, within the smallest tolerance of any measurement device on earth, a triangular structure with sides 3-4-5, all the properties derived from the axiomatic definitions will apply, but only insofar as we can measure them. We can’t measure to infinite decimals, so there is no practical way of verifying the exactness of the 3-4-5 properties other than to note that they logically follow from the axioms of geometry. So far, all triangular structures following the 3-4-5 design obey (up to our ability to measure them) the properties predicted by the axioms, so we dub the concept useful and teach it.
Lastly, there is nothing that says that what goes on in my brain has to be “real life.” Heck, a good part of the entertainment business is built on the idea that people like to see things that can’t happen in reality. It’s called imagination. The mechanics of how we imagine lie in the chemical reactions that take place between neurons; IANA neuroscientist, so I don’t have the specifics. But to claim that what goes on in one’s head has to be real because the brain is real doesn’t follow.
Keep in mind some basic definitions/properties:
e[sup]rt[/sup]: the amount a quantity multiplies by after t time-units, if its rate of change at any moment is always r (per time-unit) * its current value [that this depends only on the product rt is apparent with a little thought about the arbitrariness of the time-unit]
π: the distance halfway around a circle of radius 1
i: square root of -1
Now, let’s find another way to look at i: Suppose you were to take some vector and rotate it 90° (a quarter revolution). Now rotate it 90° again. What’ve you accomplished in total? You’ve rotated a full 180°, a half-revolution; you’ve ended up with the opposite vector from what you started with. That is, (rotate 90°) * (rotate 90°) * v amounts to the same thing as -v; i.e., in this sense, (rotate 90°)^2 = -1. That is to say, “i” essentially means “90° rotation”, nothing more esoteric than that. This is the fundamental insight.
Ok, so what? So now let’s think about what happens if you were to keep rotating a vector at a constant rate. Well, as the endpoint of that vector traces out a circle, its velocity at any moment will be perpendicular to the vector itself; that is, it will be aligned with the vector rotated 90 degrees. And, for convenience, let’s pick units of distance and time such that the vector itself has length 1, and its endpoint is always moving at a speed of 1. Thus, we will have that the endpoint’s velocity (i.e., the vector’s rate of change) at any moment is always equal to the vector itself rotated 90°.
Thus, by the defining property of e above, after t time-units of such rotation, the vector will have multiplied by e[sup](rotate 90°)*t[/sup], which is to say, by e[sup]it[/sup]. And since the endpoint is moving at speed 1 around a circle of radius 1, it will take precisely π time-units for this to go halfway around the circle (this being the defining property of π above). That is, e[sup]iπ[/sup] = 180° rotation = -1.
That’s all there is to it. When you boil down past all the defining properties, that “e[sup]iπ[/sup] = -1” simply means “If a quantity whose rate of change, per time-unit, is always its value rotated 90°, then after the number of time-units equal to the ratio between half the distance round a circle’s circumference and its radius, that quantity will have reached the negation of its original value”, just cast in slightly different words. And that this is true follows immediately from and essentially amounts to nothing more than the simple geometric fact that if I keep facing you while moving to my right, then I will trace out a circle around you.
Thank you Indistinguishable. That’s the first time I’ve ever understood how that relation could possibly make sense.
this thread is an exemplary example of a little learning being a dangerous thing. i still can’t figure out if confission is just being contrary for contrary’s sake or if he actually believes what he writes.
the pythagorean theorum is well proven.
pythagorean triples are just integer values that fit into the pythagorean theorum nicely. 3-4-5 is just the most common example.
numbers exist just as much as the units they keep track of. if you don’t believe in the number 1, what’s to stop you from losing faith in a mile, a degree, or even discrete objects like books or sheep? this conversation went from mathematical to philosophical without ever deviating from moronic.
I don’t think confission is disputing the fact that the Pythagorean Theorem can be easily proven. I think his questions here, like those he wrote for a thread on what a sperm feels, are intended to be philosophical. Heck, the Sperm thread managed to generate 83 posts out of a pretty simple question.
The point I am trying to make, and I am far from the first one to make it, is that mathematics is an abstraction and simplification of reality. It’s based on axioms and theorems that logically follow from them. When stepping into the real world, all one ever finds are very accurate–so accurate that the deviation doesn’t matter–approximations of the purely theoretical ideas of “triangle”, “circle”, “square”, etc. Sure, for all intents and purposes, we can treat these objects as if they were “perfect” in the axiomatic sense, and we do. But the OP seemed to be wanting a very literal discussion of what is real and what isn’t. Another example more dear to my heart is the idea that data we observe can come from a normal distribution. The theoretical normal has infinite tails, and I can’t think of anything real that is “infinitely small” or “infinitely large”. But chosen properly, the normal can model real data quite well, since the activity in the tails becomes arbitrarily close to 0 the further one goes out.
In high school, I wrote a program to find Pythagorean triples and eventually discovered this pattern:
[2n(n+1)][sup]2[/sup] + (2n+1)[sup]2[/sup] = [2n(n+1)+1][sup]2[/sup], for all n = N[sub]0[/sub]
This generates (where a[sup]2[/sup] + b[sup]2[/sup] = c[sup]2[/sup]):
0 1 1
4 3 5
12 5 13
24 7 25 etc.
Every odd number is a valid b-term, and a and c differ by 1. I only found out much later that other forms of this equation exist to generate other triples.
The op’s mathematical premise is not a unique one. It’s essentially Intuitionism. It could certainly be considered that L. E. J. Brouwerwas misguided, but probably not a complete moron.
Is there is the connection between consideration of the abstract 3 4 5 triangle, which does not exist in the physical world, and the consideration of supernatural to explain the source of goodness?
Don’t both atheism and intuitionism have a some connection in the form of anti-realism?
???
Only if you can show them both recapitulating phylogeny, preferably caught in the act in a seedy motel room.