non-archimedean in lay terms

Question

I've dabbled with studying infinitesimals off and on for years ... Robinson, Keisler, Bell ("Smooth Worlds"), etc., even a bit of category theory. But I'm not a mathematician and tend to jump in way over my head ( so I apologize for the large gaps in my informal training ).

The concept I keep floundering in is that of non-archimedean fields.

I understand pretty well what fields are - and I do understand the algebraic structure and ordered field concepts in archimedean fields --- it's the "non"-archimedean part I don't Grok. I am having trouble visualizing this. Well, one example of non-archimedians are infinitesimals - not exactly visualizable either (probably a math-geeky pun there).

Can someone please give an example or two of a non-archimedean structure, object, beasty - but in layman's terms ? (Yes, I have read the wiki stuff.)

[Edit] found these useful after some comments received:

Intuition behind "Non-Archimedean" -- two senses of "non-archimedean".

Example of a complete, non-archimedean ordered field

And this was a good refresher (for me at least) on ultra filters in this context: A layman's motivation for non-standard analysis and generalised limits

Also curious why an editor removed the Field-Theory tag I put on here. Non-Archimedean Fields are not considered part of Field theory ?? If not, then where's the Non-Archimedean Field Theory tag ? :-P

For a field, isn't being non-Archimedean equivalent to having infinitesimals? So if you can't visualize infinitesimals, then you won't be able to visualize non-Archimedean fields either. — Trevor Wilson, Sep 06 '13 at 22:12
If you read all that, then you probably saw the terms "Ultrapower" and "Los theorem". An ultrapower of $\Bbb R$ by a free ultrafilter on $\Bbb N$ gives birth to a non-standard field. It's quite simple to see how once you are familiar with the basics of ultraproducts. Are you? — Asaf Karagila, Sep 06 '13 at 22:16
@Asaf https://en.wikipedia.org/wiki/Archimedean_property#Non-Archimedean_ordered_field seems like an easier example for non-logicians — Trevor Wilson, Sep 06 '13 at 22:16
@Trevor: I don't know, ultrapowers are fairly simple to understand (at least if you understand what a structure is, and what is an equivalence relation). I never had an issue with that, and I always found myself struggling with all those other constructions of fields using polynomials. — Asaf Karagila, Sep 06 '13 at 22:19
@Will: Only if you like your friends to stab you in the back. Ultraproducts, on the other hand, have the term "ultra" in the name, which makes them infinitely more awesome right away. — Asaf Karagila, Sep 06 '13 at 22:27
@Asaf, I see you've been talking with Jonas again. Although I'm not sure how he feels about polynomials or ultraproducts. I put in the diophantine equation yesterday at http://en.wikipedia.org/wiki/Apollonian_gasket#Integral_Apollonian_circle_packings and they have not deleted it yet! Also, I use Ultrabrite toothpaste and am consequently a little awesome. — Will Jagy, Sep 06 '13 at 22:30
@TrevorWilson are there then any non-archimedeans which are not equivalent to infinitesimals? And (as I understand it) infinitesimals are always non-archimedean ? (See Will Jagy's answer below) sorry, the not " visualizing an infinitesimal " was a tongue-in-cheek weak reference to Berkley's Ghost of Departed Quantities. @ Asaf karagila - like I said, my training has gaps, but I do understand superficially how ultra filters generate Robinson's non-standard fields, but I don't get the non-archimedian connection. Duh ... are all non-standard fields by definition non-archimedean ? — Howard Pautz, Sep 06 '13 at 22:38
The term "non-Archimedean" applies to structures (e.g. ordered fields) and the term "infinitesimal" applies to elements of such structures, so these two terms are not interchangeable. However, "is non-Archimedean" is equivalent to "has an infinitesimal element." See https://en.wikipedia.org/wiki/Archimedean_property#Definition_for_linearly_ordered_groups. — Trevor Wilson, Sep 06 '13 at 22:55
@TrevorWilson oh very good ! There it says: "Let x and y be positive elements of a linearly ordered group G." of which some could be infinitesimals or infinite elements. And "The algebraic structure K is Archimedean if it has no infinite elements and no infinitesimal elements." I was conflating the two, considering them interchangeable... thx — Howard Pautz, Sep 06 '13 at 23:17
Note also that internal set theory, one of the major versions of nonstandard analysis, reveals "really big" numbers in $\mathbb{N}$ without adding any new elements to it, so that $\mathbb{R}$ retains the Archimedean property. (Technically there are no infinitesimals in IST, but it certainly feels like there are.) This is done by introducing three new axioms governining the use of a new predicate, "standard," which gives us a richer vocabulary for talking about particularly big and small numbers. With it, we don't need to extend the number line to get the benefits of infinitesimals. — pash, Sep 07 '13 at 15:38
@pash can you link in some references, please ? Not on ISA as that's too broad, of course, but rather on how ℕ can have "Really big" elements and the system not be *R ... I understand that's part of what Prof. E. Nelson did, but the standard predicate "feels like" a meta-level A.ofC. ( ? ) thx - very interesting... — Howard Pautz, Sep 08 '13 at 23:34
@Howard Pautz, you might take a look at Nelson's original paper. For getting practically acquainted with the axioms of IST, I recommend Alain Robert's Nonstandard Analysis. Note also that there is a "radically elementary" version of IST, which uses as axioms some facts that are theorems in full IST; you can get a feel for what it's like to use infinitesimals just looking at that—search for "radically elementary" nonstandard analysis. — pash, Sep 09 '13 at 15:16
@pash: I would like to comment on your parenthetical remark above that "Technically there are no infinitesimals in IST, but it certainly feels like there are." I think the "technical" claim is debatable. Let's take a reasonable definition of an infinitesimal as a number $dx$ such that we can calculate the derivative of $y=x^2$ by letting $dy=(x+dx)^2-x^2$, forming the quotient $dy/dx$, and taking the unique standard number which is closest to this quotient. With this definition, there are certainly infinitesimals in the IST line. — Mikhail Katz, Sep 10 '13 at 18:26
@user72694, by "technically" I mean with respect to the usual definition of "Archimedean," which is given in the Wikipedia article linked above. Of course one can argue that this definition does not capture the right idea. — pash, Sep 11 '13 at 15:08
@pash: In Nelson's framework, $\mathbb{R}$ is Archimedean (which captures everybody's intuitions alright) and yet it contains infinitesimals in the sense described above. — Mikhail Katz, Sep 11 '13 at 15:41
" ℝ is Archimedean ... w/infinitesimals ... " This is very interesting. So, Nelson's paper was in 1977. Has IST gained common / general acceptance (aside from the issues arising out of ZFC/axiom of choice) ? A simple yes, no, still debated would suffice. — Howard Pautz, Sep 11 '13 at 18:46
@user72694, by "the sense described above," you mean as in your comment, right? My point is that there are no infinitesimals in IST according to the definition of "Archimedean". Certainly IST has so-called infinitesimals in some sense, but not in that sense. This is significant because it differentiates IST from Robinson-style NSA and other "extensional" flavors of NSA. Even though it gets you the same place, IST is really fundamentally quite different. — pash, Sep 12 '13 at 14:53
@pash, I agree that they are fundamentally different, but nonetheless they are equivalent in the sense that if one replaces Nelson's "set" by Robinson's "internal set" one gets an isomorphism for a suitably saturated model of NSA. In that sense, one has infinitesimals in IST if and only if one has infinitesimals in NSA. Note that NSA satisfies the following version of the Archimedean axiom: for all $x>0$ there is a hypernatural $n$ such that $nx>1$. — Mikhail Katz, Sep 12 '13 at 15:12
@user72694, yes, that's a good way to put it. ... On the other hand, by the same reasoning one may argue that if there are no infinitesimals in IST, then there are no infinitesimals in Robinson-style NSA either. ;) — pash, Sep 13 '13 at 12:13
@pash: I think I follow everything you wrote except for the expression "on the other hand". Rather, this seems to be exactly what I was saying. — Mikhail Katz, Sep 15 '13 at 16:35
@user72694, sorry, I missed the "and only if" in your comment and just wanted to emphasize that the implication holds in both directions. I agree with what you've written. — pash, Sep 15 '13 at 19:25

score 4 · Answer 1 · answered Sep 06 '13 at 22:19

4

The cheap version is this: rational functions in one variable $x,$ where a function is called "positive" if it is eventually positive as $x$ goes to $+\infty.$ One function is greater than another if the diffference is positive.

In this field, $\frac{1}{x}$ is smaller than any positive real, yet is also positive. Therefore "infinitesimal"

NOTE: I have not read the Wiki stuff. If you wish detail in something intended as a textbook, I suggest Hartshorne's Geometry:Euclid and Beyond.

answered Sep 06 '13 at 22:19

Will Jagy

139,541

1

Trevor Wilson also linked to the Wiki stuff in the third comment, first answer: https://en.wikipedia.org/wiki/Archimedean_property#Non-Archimedean_ordered_field --- all that seems to hinge on the concept of non-linearity (?) and here's the main wiki https://en.wikipedia.org/wiki/Non-Archimedean_ordered_field. Thx for mentioning Hartshorne - looks like I'll be out of $20 soon... – Howard Pautz Sep 06 '13 at 22:46

score 1 · Answer 2 · answered Sep 07 '13 at 01:28

1

The lexicographic ordering on the points in the plane $\mathbb{R}^2$ gives a simple example of a non-archimedean ordered group. In the lexicographic ordering, up-and-down movements (i.e., changes in $y$) are insignificant (infinitesimal) compared with left-and-right movements (i.e., changes in $x$).

answered Sep 07 '13 at 01:28

Rob Arthan

48,577

Very Nice! - much closer to a lay definition I was asking for. Though I understand somewhat the more technical answers, was looking for a good visual... even considered asking for the "child's version" until I read on meta.math what would happen if I did so. Would anyone be shot down for giving the frog hops on rocks crossing the river analogy ? – Howard Pautz Sep 07 '13 at 01:33

score 1 · Answer 3 · answered Sep 10 '13 at 18:06

Since I see you are in systems assessments rather than technical pure math, I would like to propose to your attention Efthemiou's approach to infinitesimals, which may satisfy your request for a layman's definition.

The Atomic Theory of Calculus by Costas Efthimiou

Of course you have heard many times that all objects in this universe are made of molecules which are made, in turn, of atoms. Atoms are the smallest - once thought indivisible units - of matter. It comes as no surprise to you when I say that my watch is made of atoms, my hand is made of atoms, and so on. But it may come as a surprise to you if I say that all mathematics that studies the continuous and smooth changes of the physical world - that is, all math that is based on calculus - is made also of some indivisible units - the math atoms - that are called infinitesimals. And in the same way that atoms have definite rules of behavior that dictate how they may combine to form molecules and objects, infinitesimals also obey rules that dictate how they can be used to derive all mathematics. Mendeleyev discovered the order of the physical universe, while Newton and Leibnitz discovered the order of the mathematical thought. Since mathematics is the language (tool) of science, understanding calculus is essential for understanding science. And you can understand calculus and how it is applied only if you understand its building blocks, the infinitesimals.

thx for the lovely, plan-language insight. As a systems guy, math became easier for me when I realized it's basically a sophisticated programming language - one that defines its own 'behavior'. (Consider e.g. the parallels between Category theory and Object Oriented Programming. Or tensors and multi-dimensional arrays, etc. etc.) RE the atom analogy: Did you mean to say that infinitesimals are the most fundamental, foundational building block of math: "infinitesimals also obey rules that dictate how they can be used to derive all mathematics." ? If so, you've just tetrated my interest ! — Howard Pautz, Sep 10 '13 at 19:59
... (continued) ... or all of *calculus* which the last section of your answer says by extension. — Howard Pautz, Sep 10 '13 at 20:08
The adjectival form of Tetration ? :)) So, did you really mean All or calculus ? (I've reread your answer several times, but I still can't tell.) — Howard Pautz, Sep 11 '13 at 18:36
@HowardPautz: Efthimiou's view is that infinitesimals are the root to understanding all of calculus. Note that analysis is a vastly larger field than calculus. — Mikhail Katz, Sep 12 '13 at 13:10

non-archimedean in lay terms

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