The first number of the set is x, the second defined as 0. No matter how far down you go, you end up with x starting the set and every other one being zero. It matters as much as the infinite zeros to the left and right of a number.

you're acting as if ((x,0),0) is the same as (x,0), but we (Dave, etc.) already ageed that x is not the same thing as (x,0), although there is claim of isomorphism, so that cannot hold. see, it's equivocations stacked on top of equivocations. turtles all the way down!

It saddens me that discussions such as this one can take place on this forum. I suppose next we will be harangued about the state having an interest in us buying their 'lies' about the English language, and told that actually book means foot or something. Then not only will we be unable to understand math, but also unable to communicate.

you just equivocated again: multiplication has a different meaning in the original reals vs. in the reals in S, as i think you will agree. so we cannot say that "square" or :"square root" mean the same thing in both.

Nobody said they mean the same thing in both. We said that "square root" has a meaning in both [though of course different, since it is talking about different operations on different sets], and that a sentence containing the phrase square root is true about the reals if and only if it is true about the reals in S.

realistically, Dave, I don't expect you to carry on (though it'd be nice) as what i'm saying must sound crazy. that is understandable, and i'll continue to respect you as a valuable poster on these forums whatever you decide.

but right here we have finally gotten to the core of the matter, with the part in bold, and anyone else is welcome to argue with me on this.

the bolded part demonstrates exactly what i mean when i say "it is equivocations all the way down". this, gentlemen, is the bottom. the fundamental belief that a word is the same as a concept, that the word IS the thing defined, that an utterance can be true or false rather than a notion being true or false.

two identical sentences containing the phrase "square root" can be uttered in reference to both sets, and can correspond to notions in the speaker's mind that are true for each set. the problem is that these are two different notions in the speaker's mind. to point to the utterance being identical is only to embrace equivocation not as a convenient usage as suggested earlier, but as a FINAL DEFENSE. it really is equivocations all the way down.

It saddens me that discussions such as this one can take place on this forum. I suppose next we will be harangued about the state having an interest in us buying their 'lies' about the English language, and told that actually book means foot or something. Then not only will we be unable to understand math, but also unable to communicate.

Not quite, but close.

If I wrote it more than a few weeks ago, I probably hate it by now.

It saddens me that discussions such as this one can take place on this forum. I suppose next we will be harangued about the state having an interest in us buying their 'lies' about the English language, and told that actually book means foot or something. Then not only will we be unable to understand math, but also unable to communicate.

+1

All this time the state has also made us believe that A is the same as A even though the latter is actually 17 character spaces to the right of the former.

Mr. Katz, although Player has suggested what you say, i am not suggesting this. either way, discussion should be encouraged. even if Player (or me) is making a fundamental and obvious error, why is it that it saddens you to see this, when it doesn't appear to sadden anyone when a newbie comes in and makes fundamental errors in economics? any prejudice against the questioning of authority, even if that questioning appears obviously misguided, should be a red flag.

if anyone talking about these things is so wrong, they should be put in their place by argument to the point, just like when a marxist pays us a visit. (not to imply that i am wrong like a marxist, but just to acknowledge that it surely must seem like that from your perspective.)

You hit the nail on the head. Someone coming in off the street, so to speak, doesn't sadden me because he's not, yet anyway, a member of this forum. I'm not hoping he will make contributions to the field. He does not demonstrate the thinking of a good Austrian. However, people who do stand for the Austrian school are implicated here. If there were legitimate confusion, it's been answered. In essence, though, the real issue is that there are two approaches when you come across something you don't understand (there's nothing wrong with not understanding analysis, it's a hard subject) but wish to. One is to sit down and learn about it, asking questions along the way, and then evaluate what you've learned. The other is to spout off with statements like "it's all wrong anyway." I will not educate anyone here about how models and structures work, what representation means, or the incredible multitude of problems found within 5 minutes of looking at the mathis site. Why not? Because these are real subjects, worthy of actual study. That's why books exist on these topics. That's why I taught them in colleges. It does not do to attempt to sum up the methods and meanings of these fields in posts on a message board. Any attempt to do so leaves an opening (since it doesn't do justice to its field) for you to misinterpret. You have a degree in math, which means you ought to be well prepared to pick up a text in analysis, one in logic, and one in model theory. If you wish, I'm happy to give recommendations. If you have questions while you read, I'm happy to answer them. (My graduate training is in math and philosophy, my current research concerns reverse mathematics, which is the study of what axioms are necessary to prove specific theorems.) If you wish to make inane comments, I'm not interested.

again, i am not defending Miles Mathis. if i made a comment that seemed to you inane, you may ignore it or zing me on it, at your choosing. telling me to pick up a textbook is missing the point that i am saying these textbooks are built on sloppy foundations. they may work for mathematics and then it would not be fair to call them sloppy for that; but i started this thread not to talk only about pure math but about the whole of math and science.

physics is the modern kingmaker, and where most of the action is, so as a preview i will say that this will usually tunnel down into some problem with modern physics. what i am saying will either seem wrong or frivolous until that context is established, but i am willing to take that risk.

Let's be specific, then: what is sloppy about the foundations? Now, it won't do to point me to well-known theorems in foundations, those aren't sloppiness but known limitations on axiomatization. Nothing you've done here has gone to foundations; what you've talked about here is the kind of thing that can be learned from textbooks because it does not go to foundations.

Now, I don't know what you mean by "the whole of math and science." I'm also not a physicist, but my understanding is that math is the language of science. That is, the axioms used in mathematics are not claimed to be in some way 'true of the world.' Rather, they establish a basis for studying patterns, and once we've identified a pattern in the real world, the abstract studies allow us to say things about it. If we establish that any pattern having X has also Y and never has Z, and we come across something with a pattern that has X, we know it has Y and not Z. If you want to say that, in fact, it doesn't have X, then you're not making a mathematical statement. If you wish to challenge the claim that X implies Y and ~Z, then you might be asking a question that has to do with the foundations of math - it could also be a simple mistake in the deduction.

We don't need some sort of correspondence between real analysis and the real world. We need only (and this varies for the application) an idea of what we're abstracting on, and the assurance that we have a good abstraction of that thing. Statistics works on the real analysis framework, and so does calculus, but in entirely different ways.

i agree with everything you wrote there. the question is how to properly define the assignments of mathematical abstractions to physical objects and relations, and that is what i do not think is done properly.

but as regards pure mathematics, i am not too happy with how it takes even the natural numbers as givens (or how it defines them when it does).

i suppose this is hard to attack because mathematics can be defined to encompass so many different things. if we speak of numbers at all, in a sense we are stepping out of pure formalisms and into some semblance of physical worldliness.

let me put it this way: is there such a thing as pure formalistic math that it is possible to comprehend completely without reference to anything experiential? my answer is no, but i would like to hear others.

Mathis is claiming that the "+" symbol is misleading in the z = x + yi definition, because normal addition has no allowance for adding a real number to an ordered pair. how does one add a real number x to an ordered pair (0,y)? the obvious answer is that the real number was not actually a real number all along, but itself an ordered pair: the ordered pair (x,0).

Any standard definition states that a complex number z=x+yi is the sum of a real part and an imaginary part. Strictly speaking, it is not defined as the sum of a real number and a complex number; although this can be commonly found. Rather, it is the sum of a complex number (although one in the subset S, or ordered pairs on the real line) and a complex number, two perfectly conformable elements. In short it can be written as: z=(x,y)=(x,0)+(0,y)=(x,0)+(y,0)(0,1).

The last step is then: (x,0)+(y,0)(0,1)=x+yi (i.e. x=(x,0) and y=(y,0)); the step which you have the problem with, and which implies z=x+yi. Usually this step is not proven rigorously (that is, that x=(x,0), y=(y,0)). But this can be shown by proving that there exists a unique mapping back and forth (isomorphism) from each element in R to each element in S (ordered pairs on the real line). Hence, the equality is legitimate. In short, we can use (x,0)+(y,0)(0,1)=x+yi, because every operation we perform on x+yi preserves everything as it would be if instead we always worked with (x,y) or (x,0)+(y,0)(0,1) or (x,0)+(0,y). Thus, we can find (x,y)(u,v) by instead finding (x+yi)(u+vi) and then converting correspondingly to get the final ordered pair; NOTHING IS LOST.

but for that matter "adding a real number and an imaginary number" cannot be talking about normal addition. what is 5 plus a donut? we could define such "addition" but it would not be the same as the addition we are accustomed to.

Yes, modern physics has become a neo-scholasticism. It is the avoidance of real questions in the pursuit of trivial methodology. It is the memorization of an endless list of names and manipulations in lieu of understanding mechanics. It is the setting up in some black data hole and extemporizing on an endless string of evermore ridiculous hypotheses instead of looking at known physical problems closer at hand. It is the knee-jerk invocation of authority and the explicit squelching of dissent. It is the hiding behind tall gates and a million gatekeepers, and euphemizing it as "peer review." It is the institutionalized acceptance of censorship and the creation of dogma. Grand Masters like Feynman say "shut up and calculate!" and everyone finds this amusing. No one finds it a clear instance of fascism and oppression. An internet search on "against Feynman" or "Feynman was wrong" or "disagree with Feynman" turns up nothing. The field is monolithic. It is completely controlled and one-dimensional. All discussion has been purged from the standard model, and all debate has been marginalized. Any non-standard opinion must be from a "crank" and blacklisting is widespread. Publishing is also controlled, both in academia and in the mainstream. Einstein already found science publishing too controlled for his taste in the 30's, refusing to work with Physical Review. What would he think now? Can anyone imagine his early papers getting published in the current atmosphere?

QFT. My brother has a theoretical Physics Masters and describes the current state of physics similarly. Rarely do they attempt to grapple with the fundemental issues but assume lots of equations, in particular Maxwell's, as given and just keep turning the handel of the black boxes.

I am not endorsing Mathis, I know nothing about him, however his above analysis seems bang on.

The atoms tell the atoms so, for I never was or will but atoms forevermore be.

That is not my subject, except to point out that, in fact, we don't necessarily do that. We don't assign mathematical objects (which I take to be what you mean by abstractions) to physical things, necessarily, and then let the relations follow.

Which definition of the natural numbers are you referring to? In standard approaches it is either done as 0=null, 1={null}={0}, 2={0,1}... or as 0=null, 1={0}, 2={{0}},.. Which one are you referring to, and what makes it problematic?

As for your final question, the answer is no, as is known after Godel's theorem, if I'm understanding the question correctly.

My friend, you've taken basic courses in algebra if you have a first degree in math. So you know that + is used to stand for any abelian binary operation. However, whenever we use + in R^n, we have what Dave mentioned - that is, isomorphism. + in C is the elementary extension of + in R.

The last step is then: (x,0)+(y,0)(0,1)=x+yi (i.e. x=(x,0) and y=(y,0)); the step which you have the problem with, and which implies z=x+yi. Usually this step is not proven rigorously (that is, that x=(x,0), y=(y,0)). But this can be shown by proving that there exists a unique mapping back and forth (isomorphism) from each element in R to each element in S (ordered pairs on the real line). Hence, the equality is legitimate.

this is just a new equivocation: now the "=" refers to isomorphism rather than standard real number equality.

JAlanKatz, this is all going to come down to "what are (natural) numbers"? if we take it for granted that "+" can stand for any abelian binary operation, we have sidestepped this issue already.

i think of the natural numbers as either movies of counting or sets of sensory objects. it we define them in the set-theoretic way you are talking about with 0=null and such, are not we really making an abstract mathematical object that has nothing to do with 2 donuts on my dinner table (rather than {0,1} donuts)?

Zangelbert Bingledack, that is irrelevant. If we call the set of real numbers with usual addition and multiplication R and define the set of complex numbers C as pairs of real numbers (a,b) with a,b in R with addition defined as (a,b)+(c,d)=(a+c,b+d) and multiplication as (a,b)*(c,d)=(ac-bd,ad+bc), the statement "The subset R' defined as { all x in C with x=(a,0) where a is in R } is isomorphic to R" merely states that it is completely irrelevant where you execute a given calculation, whether in R or R' since we have a one-to-one correspondence in elements and calculations. It does not state that R and R' exactly the same as in A=A but they do not differ in any meaningful way, that is in the way calculation with their respective elements are made.

You see, there is no contradiction. You construct somehow, there are a few equivalent ways of doing this, a body R, construct with it a body C, show that a subset of C behaves exactly like R, call this subset R' and do all your calculations in this new body - or your old R, since it does not matter.

Honeste vivere, nemimen laedere, suum cuique tribuere.

"...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 a*0 := 0 and a*S(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.

Honeste vivere, nemimen laedere, suum cuique tribuere.

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.

Honeste vivere, nemimen laedere, suum cuique tribuere.

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....

"And it may be said with strict accuracy, that the taste a man may show for absolute government bears an exact ratio to the contempt he may profess for his countrymen." - de Tocqueville