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.
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.
Not quite, but close.
+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.
Z.
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.
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.
I edited my response; there were some errors initially.
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.
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.
deleted - wrong thread
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.
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.