Talk:Foundational status of arithmetic
|
|
Yow! There are far too many bold words on this page! RickK 03:47, 14 Sep 2003 (UTC)
It also have virtually no web presence: generatics (http://www.google.com/search?q=generatics). The term generatic (http://www.google.com/search?q=generatic) seems to have some kind of established usage, but this isn't it. -- Cyan 03:59, 14 Sep 2003 (UTC)
- I got rid of the bold. Can't comment on the content - not in a mood for reading! Angela 04:06, Sep 14, 2003 (UTC)
I never heard the word "generatics" before seeing this article. Of course everyone (except perhaps non-mathematicians) is familiar with both
- the characterization of mathematical objects by axioms (e.g., the real numbers from a complete ordered field), and
- the characterization of mathematical objects by constructing them from more primitive mathematical objects (e.g., the real numbers are equivalence classes of Cauchy sequences, or are Dedekind cuts, etc.). The latter seems to be what is meant by "generatics".
Cyan comments on "web presence" above; can anyone tell us if this is found in the literature somewhere? Michael Hardy 20:27, 14 Sep 2003 (UTC)
At the beginning of the article, I cite the comment from Morris Kline's history about Russell's indirect praise of the "genetic" nethod over the "axiomatic metod". (Apparently without the "bold", this clear statement seems overlooked by some of the above commentators. Also seemngly overlooked is my "point" that generatics parallels the way we learn language, so that what one learns in language can help in learning arithmetic, and vice versa.) I've often encountered this foundational term, "genetic", elsewhere in the literature. But it sounds biological. So I formulated "generatics", from "generate", to avoid bio-confusion and to resemble the term "axiomatics", which is found throughout the literature. I'm willing to compromise in any agreeable way. Michael Hardy seems to favor adjoining an uncountable infinity of irrationals to avoid a methodology which easily extends from a methodology familiar in pedestrian statistics and arithmetic. As noted above, the two standard ways of founding reals is by Cauchy sequences and Dedekind cuts. I like Dedekind cuts, but a neophyte might look for this at the butcher counter, whereas "decimals" are known to any one with a little secondary math. You need only explain (as I do in the article) that a "rational decimal" has a repeating digital segment, but a "irrational decimal" does not. ONLINE, hyperlinked from the "redux" listing in my "Refernces", I've a file easily showing how to turn a rational decimal into an irrational one. Take 1/9 = 0.1111..., which repeats the 1-digit infinitely. After first 1, write 0; after second 1, write 00; ...; after nth 1, write n zeros; etc. Obviously, this doesn't have a repeatable segment, hence, is, by definition, irrational. That I explain it in a finite number of words shows it can be mathematically explicated. Johnhays0
The concern here is that this article represents may represent an original synthesis on your part, Jonhays0. Among the things that Wikipedia articles are not supposed to be are personal essays and primary research. It's not at all clear at this point that the generatics article is neither of these things. That doesn't mean that the material you've written doesn't have a home in Wikipedia - but this article may not be it. -- Cyan 23:37, 16 Sep 2003 (UTC)
An email Jonhays0 sent to Cyan:
It did not begin with me. The generation by Hamilton of complex numbers from vectors of reals can be found in many books. I learned this procedure in 1957 as the result of organizing the first National Science Foundation Institute in Puerto Rico -- for high school math teachers of the Island and from the States. Before that Institute began, the NSF sent me printouts from previous Institutes. One "memoir" told about Hamilton and how to derive integers as vectors of natural numbers; and rationals as vectors of integers. In the Institute, I gave a series of lectures on "The Foundations of Mathematics" and taught this procedure. Many able students discovered for themselves the "rule of signs" as whatever preserved the definedness of operations on defined differences of naturals; the rule for dividing a fraction by a fraction as whatever obtained a defined quotient of integers from dividing a defined quotient of integers by a defined quotient of integers. This explictly does what is implied in the famous statement about "invariance of form" of 19th century mathematician, George Peacock. You may quarrel with my presentation, but you cannot document your statement that this began entirely with me. --User:Jonhays0
I did not mean to make such a claim (although that is certainly one way my original comment can be interpreted). I apologize for not communicating more clearly, and I have amended my statement above. I don't wish to quarrel at all; what I want is to improve Wikipedia, in accordance with the generally-agreed-upon guidelines and policies.
Here is the original synthesis that I alluded to earlier: I believe that you have gathered these similar strands of the philosphy of mathematics together, and placed them under the novel title "Generatics". This may be a problem because most professional mathematicians have never heard of this term; when they see it in Wikipedia, they are likely to conclude that it is a crank article (which wouldn't do justice to your article at all). (For an example of a fairly crankish article, see Transcendentalist Hypothesis.) The net result would be that Wikipedia's reputation as an authoritative source of information would be damaged. It can also be a problem in that Wikipedia readers aren't likely to search for the term "Generatics", so your article will languish unread, which again wouldn't do it justice.
So what's the solution? Well, it may be that a good place for the material is in Philosophy of mathematics. If it is placed there, it will have better exposure, and it won't set off any false alarms about idiosyncratic views. If you choose to move it there, it will need some redacting in order to fit in with the material that is already there. I suggest that you avoid the specific term "Generatics"; you can easily explain all the essential genetic/generative ideas without using it.
I hope this rather long post has clarified my concerns. -- Cyan 05:59, 17 Sep 2003 (UTC)
I realized just now that I have been referring to the Generatics article as "your" article. I want to clarify: it isn't yours anymore. Now that you've put it in Wikipedia, anyone at all can come and change (hopefully improve) it; this is the point of the wiki process. So I hope you haven't become too attached to its present form. -- Cyan 06:08, 17 Sep 2003 (UTC)
I propose to revise this article in the way suggested by Cyan. I suggest a new title such as Generating arithmetic; have lead-off about people having different philosophical attutudes about math, especially arithmetic. Link that to the Hilbert-Russell item. Go with basics of the article, but remove the controversial casting of "generatics vs. axiomatics". Unless y'all have objections, I'll start working on this, locally, for "uploading" by "Copy". Just how to do the "redacting", I beg for help. I welcome any inputs. jonhays 01:56, 18 Sep 2003 (UTC)
Sounds good. (Redaction (http://dictionary.reference.com/search?q=redaction) is just a fancy word for editing/revising. Sorry about that.) -- Cyan 04:53, 18 Sep 2003 (UTC)
Please explain all of the bolded phrases. RickK 03:16, 18 Sep 2003 (UTC)
RickK: As I explained to Angela, I tend to write as I lecture, raising my voice and enunciating carefully at each important phrase. I agreed with her that this is a questionable habit. And I think Angela removed many of the offenses. However, I noticed, after this, that some commentators overlooked some of my points, saying or implying that I did not do what I explictly did. I'll happily consider any further problems.jonhays 19:29, 18 Sep 2003 (UTC)
In my copy of Boyer's "A History of Mathematics", chapter 25 is entitled just plain "Analysis". It's a second edition, though. -- Cyan 04:09, 7 Oct 2003 (UTC)
I've applied a surgeon's knife to this page, to keep down the POV parts and retain the more interesting arguments. That (I do think) was required for it to have a long-term future. It needs more work, too.
Here is the bulk of what I edited out:
'(The begat operation has been taught to primary school children to prepare for later teaching of the more profound successor function.)
Furthermore, the generative methodology, in the form of recursion, is also as ancient as the gnomon used by Pythagoras and the [Pythagoreans]] to label rpresentations of the unit of a basis to generate number patterns and enable their comprehension, as shown in Figurate numbers. The generalization of a basis unit is indicated in the online Wolfram mathworld: "Speaking in general terms, an object is 'generated' by a basis in any manner that may seem appropriate."
This type of derivation has been labeled as the "genetic" methodology in the foundations of mathematics in a famous retort of British mathematician, logician, and philoopher, Bertrand Russell. Countering the claim of David Hilbert for the superiority of the axiomatic method, mathematician and mathematical historian, Morris Kline, paraphrases Russell as saying that the axiomatic method has "the advantage of theft over honest toil. It assumes at once what can be built up from a much smaller set of axioms by deductive arguments."
The mathematical legitimacy of generating arithmetic is doubly founded.
One foundational "plank" is "the most sacred rule of mathematics", namely, equivalence. (In Euclidean geometry, the primary concept is congruence, which is an equivalence. Other forms of equivalence abound in mathematics.)
An equivalence rule E on a system S reformulates S into equivalence classes. In science, this is the mathematics behind biological formulation in species, genera, classes, etc.; behind atom and molecule and element in chemistry; etc.
In mathematics, the equivalence class property is related to the ("all in the family") property of closure, as in, "The natural numbers are closed under the operation of addition." And this statement invokes the totalness property of equivalence, as in "addition is a total operation in the natural number system". In contrast, "subtraction is a partial operation in natural numbers", failing whenever subtrahend is greater than minuend. (Students learn a variant of this in being told that "two plus two is four", not "two plus two are four". The first form closes "two plus two" as equivalent to "four",
confirming the totalness of addition.)
It also leads to the group concept, one of the most vital in all mathematics:
- A system closed under an operation is a groupoid;
- a groupoid with associativity for its operation is a semigroup;
- a semigroup with an identity element is a monoid;
- >a monoid with an inverse for its operation is a group.
(A group is the mathematics underlying laws of science.)
Similar extensions of equivalence provide for generating the other number systems, erecting each in terms of vector components from an already founded number system. That is, arithmetic rules for system Si+1 derive from those arithmetic rules in system Si which (as can be disccovered) conserve equivalence.
The second foundational "plank" might be heuristically stated as follows. A formulation is mathematical if it can be embodied in a program embedded in a device (or computer simulation thereof) that (consensically) allows all trained users to enter, correctly, the same inputs into the device and obtain the same outputs.
This heuristic attains mathematical status by the theorems of recursive function theory in recursion. Among different effective methodologies in computation, a procedure is shown to be an algorithm by specifying a Turing machine which yields the same result.
- Double application of the above definition yields another equivalence: a − b = c − d if, and only if, a + d = b + c, again subtraction in terms of addition.
- This then yields, for natural numbers as vector components, the equivalence: [p, q] - [r, s] = [p + s, r + q].
And that last eqivalence implies that, for example, [2, 0] - [3, 0] = [2 + 0, 3 + 0] = [2,3] = [2 - 2, 3 - 2] = [0, 1]. That is, [2, 0] - [3,0] = [0, 1], whereby the last pair can be labeled "a negative pair". This means that, whereas 2 - 3 has no meaning in natural numbers, [2, 0] - [3,0] = [0, 1] has meaning in pairs of natural numbers since it involves only addition of naturals, a totalness operation. And [0,1] can be be defined as the integer -1.
Thus, equivalence allows generating integers as ordered pairs of natural numbers, because equivalence separates vectors of naturals into exactly three equivalence classes, prototyped as [n, 0], [0, m], [0,0]. respectively defined as positive integers, negative integers, and zero, i.e. [n,0] = +n. [0,m] = -m, [0, 0] = 0.
This establishes a one-one correspondence or equivalence between integer arithmetic and natural number vector arithmetic. The former does arithmetic, the latter discovers and explains arithmetic.
Similarly, equivalence on quotients of integers establishes three equivalence classes, prototyped as [n, 1]. [1,n], [n,m), for n > 1, m > 1, which, respectively correspond to integral rationals, Egyptian fractions, and fractions.
The various vector formalisms that generating the "beyond-natural" numbers systems of arithmetic then secure mathematical status by embodiment in online JavaScript programs (equivalents of Turing machines) wherein vector operations consensically yield the same results as their more familiar arithmetic forms.
The axiomatic procedure does not explain the properties of arithmetic as generated from basic properties with informal aspects in daily life, but rather as postulated rules. This has invoked a "School of Social Constructivism" (with many online websites) which argues that any mathematical system is merely a social construction -- on par with table manners.
The previous paragraph states the advantage of generating arithmetic. In turn, the importance of generating arithmetic is implicit in the following declarations.
Charles Matthews 10:56, 21 Oct 2003 (UTC)
I protest! I initiated this article to show young people and nonmathematician adults that the rules of airhmnetic (a model for the rest of math) can be understood. By these methods, I've seen 13-year-old students discover the "law of signs" for themselves, by requiring CLOSURE on DEFINED DIFFERENCES (DDs) of natural number: a - b, s.t. subtrahend is not greater than minued, so that difference is a natural number. Yet some one edited this article to make it appear that one can only understand CLOSURE by studying the veddy advanced math of category theory. It is the mathematical equivalent of the birds-and-bees statement that "humans reproduce humans". In arithmetic, operations on numbers of a system are supposed to reproduce numbers of the same kind of system. What's advanced about that? Please read the book, "Learn from the Masters", Eds. Swetz, Fauvel, Bekken, Johansson, Katz (Mathematical Association of America, 1991) on p. 260: "For Galois (1830), Jordan (1870), and even Klein's "Lectures on the Icosohedron" (1894), groups were defined by the one axiom of closure. The other axioms wre implicit in the context of their discussions -- finite groups of transformations." So, CLOSURE goes back at least to 1830, not to category theory (which I've taught!) developed in the last quarter of the 20th century.--This attempt to make math and science recondite, or the theology of a priesthood, inspired the late Carl Sagan to protest this "recipe for disaster". (See, http://members.fortunecity.com/jonhays/sagan.htm.) Sagan may have reacted to those who rejected him many times from The Academy of Science because of his popularization of science. This may explain also a remark of Hungarian mathematician Cornelius Lanczos, co-worker of Albert Einstein during the period descibed in Lanczos's 1974 book, "The Einstein Decade (1905-1915)". Of recondite writing, Lanczos said: "Many of the scientifc treatises of today are formulated in a half-mystical language ... of a superman."-- In the cited MAA book, you see that "generatics" did not begin with me in content, only in name. On p. 286, please find, "It was not until 1894 that J. Tannery [Jules Tannery (1818-1910), see ONLINE] introduced the arithmetic of rationals as pairs [vectors] of integers." jonhays 21:47, 11 Dec 2003 (UTC)
I edited the article. I think you'll find there is no Wikipedia recognition of protests of this kind. If you can improve the article, please do so.
Charles Matthews 21:54, 11 Dec 2003 (UTC)
cleanup
This article contains very little info embedded in very convoluted sentences. I have half a mind to put it up for deletion, but we mustn't be too rash. -MarSch 14:27, 19 Apr 2005 (UTC)