“…it is completely irrelevant where you execute a given calculation…”
you’re acting as if these calculations are one and the same; that is only true if, as JAlanKatz implied, you define the operations to be fully generalized from the start. but doing that means you are no longer talking about the real numbers people are familiar with, but were instead talking about a complete mathematical abstraction all along.
for that reason, it comes down to “what are numbers?” are they pure abstractions or are their experientially perceivable sensory objects or movies such as 4 watermelons or 3.71 kilometers traveled? they cannot be both.
Zangelbert Bingledack, you are jumping conclusions. Let us break it down. I will try to get to your question from every way possible.
It does not make a difference if natural numbers are real or not, at least not as long we are only concerned with how to construct something that satisfies our demands from anything we would call natural numbers. Historically natural numbers were natural exactly because people believed to be natural, you can count them with your fingers, they appear in nature. One apple, two apples, twenty apples, an apple tree. But that is not what I am concerned about, even if we knew that natural numbers were real we would want to know what properties they have. For this we give certain axioms, properties that we inherently expect from natural numbers. Those will be the Peano axioms, if you have not heard of them, look them up at Wikipedia.
We now assume that there is such a set N which elements satisfy theese demands and define an addition recursively for any a,b in N, with S(b) being the sucessor of b, as a + 0 := a and a + S(b) := S(a + b). Similarily we define a multiplication a0 := 0 and aS(b) = a + a*b. Can we agree on that? Now we construct a set Z where all of the elements of N are and additionally all elements a of N as -a. We then extend the the multiplication and addition analogously. Now this addition and multiplication are already not the same as in N and the elements are not too. What do you think?
Please keep your answers and questions short, I can then understand it better.
this is all very well and good, but the context of the original discussion about complex numbers was Miles Mathis’s claim that non-Euclidean geometry is used to cheat in physical theory via the equivocations present in dealing with complex numbers. as long as we are clear that these operations are NOT the same “+” and “=” operations that we started out with, there is no problem.
as an aside, looking at them for the first time in a long while, it is curious that the first 4 or so Peano axioms are actually just definitions of the symbols used. this suggests that pure formalism is the order of the day, where only the words and symbols need to align, not the actual concepts (as Dave implied earlier).
Euclidian geometry or non-Euclidian geometry does have nothing to do with complex numbers. I do not see any equivocations in complex numbers that are in any way ambigous. If you think I miss something please point as directly to it as possible. I do not see the benefit of stating that “+” and “=” on complex numbers are not the same as on real numbers.
This pure formalism comes from reductionism. From that point of view mathematics has literally no meaning. All mathematics is certain patterns of symbols, a formal language with no objects to describe.
There is no ‘standard’ sense of equality that exceeds “isomorphism on the same set.”
Another way to look at what you’re complaining about: If I talk about "the real number z " I’ve told you two things - that I’m referring to a real number, and it’s value. The first bit of information tells you that I’m looking at the real line, not a plane. On the other hand, if I refer to the complex number z, I’m looking at a plane. The plane is constructed as the cross product of two lines - one of which is precisely the same line I was looking at before. So why was it sufficient before to refer to a point as 5, and now I refer to it as 5+0i? Because if we’re looking at a line, we need less information to find the point than we do when we’re looking at a plane.
Suppose we are to meet for lunch. I can say “let’s meet on the earth’s surface, at Broadway and 48th.” I’ve specified that the universe of interest, so to speak, is the earth’s surface (a sphere, which is just a plane plus a point, so let’s exclude the north pole and we’re talking about a plane), and so I need to give two coordinates. On the other hand, I can convey the exact same information by instead saying “Let’s meet at Broadway and 48th, on the ground floor.” Now I’ve given three coordinates to name the same point, since I had not already specified that we are on the earth’s surface, so we’re working in a 3-d space now. It’s still the same point, it just has different names. This is actually the opposite of equivocation.
Since the point of math is abstraction, I’m not sure what you’re asking here. Are you asking what the standard definition of the natural numbers has to do with the real world, so we’re back at how to set up relationships? If so, here’s the motivation: we want mathematical objects that exemplify everything we know about the natural numbers. That is, whatever we know to be true ought to be ‘programmed’ into the definition in some way. In light of what we plan to do, the definition also ought to be recursive, and extend in some natural way to larger spaces. Also, 0=null is quite natural, and would be what we’d do in any definition, since it captures so well what 0 is all about. So, does this definition model all that we want the numbers to do? Clearly it does, using Cartesian product in place of multiplication (given numbers a and b with associated sets hat{a} and hat{b}, we have that the cardinality of the Cartesian product equals a times b) and a suitably defined union in place of addition. To demand that the mathematical representation of a number in some way look like the number is to forget what mathematical representations are supposed to do. As far as your donuts are concerned, they are in a 1-1 correspondence with the elements of 2 (since 2 has exactly 2 elements.) What else would you like to do with them?
This is just a way of ignoring the notion, explained a few times here, of elementary agreement. If a is contained in b, and I have an operation defined on a, the extension of that of operation to b will be such that, if the domain is restricted to a, it becomes the original operation. This is not a trivial requirement. Somehow, you’ve decided that I’m simply starting with a different operation altogether, but a different operation than what? Commonsense understandings are vague and imprecise. If you wish to do math, you need to capture the essence of that understanding in a precise way.
First year math and science student here. Let me offer my input. Much of the educational establishment today is geared towards conferring accreditation rather than knowledge. The corporate-government complex needs more highly trained, skilled drones to run its machines and staff its bureaucracy. The problem with math and science isn’t due to flawed information, it’s because most students don’t give a shit. It’s not cool to be intelligent. It’s not hip to care. The majority just sit through the boring lecture, bolt out the room once the pedant stops speaking and cram for a passing grade via online study aids. The educational ciricullum reflects this. It doesn’t teach the underlying significance or meaning of what we learn. We are just given a stream of facts and data which we’re forced to accept at face value at an extremely fast pace. The student isn’t given enough time to possibly “think” independently about what he obtains, merely to absorb enough skill to answer the test questions. This continues all the way to graduation. Everyone is obsessed with money, all that they care for is what job they get once they’re out of the educational joint. The university a pathway to a job and a higher social standing. This is the primary objective of most students. There is no joy of learning, no desire for knowledge, no passion for discovery. University is nothing but a place for formal accreditation, where the individual is made into a skilled expert in a field he knows nothing about.
Children who enter university exit 4-6 years from then still children who at best end up finding themselves needing to re-learn everything for real this time around.
By the way I need to do my homework. I’ve been procrastinating on this site way too long. You guys make me stay up all night.
Very good post, Thisprogramhasnotbeenrated. I admit that I am guilty of that type of laziness from time to time, but maybe that’s because I don’t get enough sleep for classes .
The state university system is a wreck. Nobody learns anything the way they set it up, not to mention all of the wasteful classes that one has to take in order to get the accredation. Just think of how much more efficient schooling would be on a total market…