I What really is the meaning the verb "induces," in math?

1. Oct 7, 2015

Junius

The verb "to induce" is used in numerous contexts in mathematics: maps, topologies, etc. In virtually every case, the author seems to take for granted that the reader knows what it means. Sometimes the usage is in a sentence of the form "An X induces a Y by.../in the form of .../etc.," but this is a description of the consequences of a particular instantiation of (this sort of) induction, not a clarification of what it is to "induce" something in general.

Online searches for a definition seem always to lead either to definitions of mathematical induction (the proof technique) or to uses of the verb "induces" rather than definitions of it. Can anyone provide a definition and/or provide a specific reference to one? Thanks.

2. Oct 7, 2015

Staff: Mentor

The definitions I saw define the term and give an example of the word being used.
In mathematics, induce usually means "bring about or give rise to."

3. Oct 8, 2015

Junius

Thanks, Mark, for your reply. I agree that the words you suggest express a sort of feeling that “induce” is plausibly intended to convey, but I’ve never encountered a stand-alone definition of “induce” in a mathematical context. Moreover, a couple of the words or phrases you mention cannot accurately apply in mathematics.

All the uses of “induce” I’ve encountered to date have left the word’s meaning implicit. E.g., Pedoe’s “Geometry,” which is intended for high school teachers of geometry, uses the word for the first time in a problem: “Where are the new coordinate axes which induce the coordinate transformations (i) Z = z-bar, (ii) Z = i z-bar, (iii) Z = -iz + 1?” (where ‘z-bar’ is the complex conjugate of z; @ 166). Here are a few more examples, all from Dover books that aren’t graduate level, and in each case either the first use in the book in question, or a clear attempt to suggest the meaning of “induce”:

“Let X and Y be topological spaces and f:Y->X be a function which is not necessarily continuous. The function f induces a function f’:Y->f(Y) which agrees with f and is onto.” [Mendelson, Introduction to Topology, @96].

“We recall that we have obtained the first fundamental form of a surface by means of the metric of the embedding Euclidean space. In this sense the metric of the space ‘_induces_’ a metric on the surface.” [Kreyzig, Differential Geometry, @179 — the quotes and italics make it clear that he’s attempting to give some idea of the meaning of the word, but it’s not made explicit.]

“Let {A_n, f_n} and {B_n, g_n} be two inverse limit sequences of spaces. A mapping Φ: (A_n, f_n} -> {B_n, g_n} is a collection {φ_n} of continuous mappings φ_n:A_n -> B_n satisfying the condition g_n•φ_n = φ_(n-1)•f_n, n≥1. This condition may be given by saying that we have _commutativity in the diagram_ below: [diagram]. This means that we may pass from A_n to B_(n-1) in two ways but the result is the same. Such a mapping Φ _induces_ a mapping φ:A_∞ -> B_∞ of the inverse limit spaces as follows. For each point a = (a_0, a_1. …) in A_∞, let φ(a) = (φ_0(a), φ_1(a), … ) That φ(a) is indeed a point of B_∞ follows immediately from the equations: [etc.]….”.” [Hocking and Young, Topology, @224 — again, emphasis in the original].

Maybe this is the sort of thing you mean by “The definitions I saw define the term and give an example of the word being used." Actually, though, that’s what I meant about specific instantiations — what I’m asking about is an explicit, stand-alone definition.

Also, while notions like “cause,” “bring about, “give rise to,” might be the sort of mental image we supply when we run across the word “induce,” I’d suggest that these words can’t be appropriate to the meaning of “induce” as used in mathematics. The reason is that mathematics lacks causality — mathematical statements don’t describe events. They either define abstract objects, make declarations about those objects (postulates), or are are logical statements about those objects and their relationships.

I think a plausible meaning of “induces” might be “allows one to infer” or “allows one to posit” — that seems to work correctly if you plug it into each of the examples I quoted above. But I’m just guessing — I’ve never seen an author take the trouble to define “induce” explicitly. If anyone can give a quote and a reference, I’d appreciate it!

4. Oct 8, 2015

Staff: Mentor

IMO, you're splitting hairs unnecessarily. In your first example, from a geometry text by Pedoe, changing the coordinate axes results in a change in variables. This change is effected by someone, not by the mathematical entities themselves. For that reason, "cause", "bring about", etc. are all appropriate here.
No author of a book on mathematics bothers to define every single word he uses. They assume that the reader has enough knowledge of whatever language the book is written in to be able to understand what is being said.

5. Oct 8, 2015

Junius

Thanks again. However, your point about agency in effecting the transformation unfortunately isn't pertinent. The grammatical subject of the verb "induces" is an abstraction in each of the examples quoted: axes, function, metric, mapping.

As to your second point, yes, of course no one defines every word he or she uses. However, again this isn't so pertinent to the issue I raised. My question isn't based on the notion that everyone ought to define the term, but rather that *no one does* define it, at least explicitly. Moreover, the fact that the word is italicised in two of the excerpts I quote suggests that the author in each case felt that there was something marked -- i.e. out of the ordinary, something different from what the reader might be expecting -- about the use of the word in context.

A check of a couple of linguistic corpora (in this case, Corpus of Contemporary American English (COCA) and the British National Corpus (BNC)) sheds more light on what knowledge a typical reader might have about the meaning of "induce." In both corpora, most uses of the word fall into three categories: (i) where what is being induced is a psychological state, (ii) where what is being induced is a medical condition, biochemical or physiological reaction, etc. in response to a stimulus, and (iii) where what is being induced is a behaviour by human agents (even where expressed impersonally, e.g. something "induces investment”).

Somewhat less often the word is used (iv) as a term of art in meteorology (or more generally, referring to motion of fluids), (v) as a term of art in electromagnetism (a 19th Century metaphorical extension of (iv)) and in other areas of physics, referring to motion (e.g., from 1992: “Jupiter's gravity, for example, induces a 13-meter-per-second solar motion”), and (vi) in the mathematical usage, which is quite rare in the corpora.

Obviously, the mathematical usage doesn’t relate to motion of fluids, even metaphorically; even less so does it refer to a psychological or physiological state or a behavior. In mathematics, some typical objects of the verb are topology, mapping, function, metric, axes, coordinate system, orientation, projection, collineation, representations, derivation, etc.

From the evidence I’ve presented above, this is a *highly idiosyncratic* use of the term in English — and yet it seems never to be explicitly defined, as far as I've seen. My question, then, isn’t about splitting hairs, but about rigor.

6. Oct 8, 2015

WWGD

I have to agree to same degree with Junius here; this is a term used (and misused) so often that it seemss right to define it more clearly. " Induce" is often used in the sense/area of category theory or sometimes algebra. There is a standard map induced in algebra when you deal with quotients ( I'll look it up and post it later, too tired now to think clearly). There are also the functorial induced maps when, e.g., a continuous map between topological spaces " induces" a homomorphism between their respective : fundamental groups, homology groups, etc. I think this may be related to naturality of maps.

Then some other times it is used more as in everyday English , to mean " give rise to". At least author should specify which of the two usages are being made for the word.

7. Oct 8, 2015

Staff: Mentor

Why is this so important? The important point is that, for example, changes in the coordinate axes result in changes to variables. What is not important is who (or what agent) causes these changes. Focusing on this minor point does not consititute rigor, IMO.
I doubt that these references would be useful in the context of fairly advanced mathematics texts. None of the three categories you listed captures the sense of the word "induces" as it would be used in mathematics. Furthermore, would a "typical reader" understand the concepts of "topological space" or "metric space"? I think this is unlikely in the extreme.
Which (item vi) backs up my conjecture that the "typical reader" would not be familiar with the mathematics concepts in which "induces" might be used.

8. Oct 9, 2015

Junius

Mark, thanks for your input. I entirely agree that the usual uses of "induces" don't illuminate the mathematical meaning -- that was precisely my point. However, the linguistic inquiry was in response to your observation
Obviously, as you yourself now argue, knowledge of English (the language the pertinent books were all written in) is NOT sufficient. So don't blame me for not understanding.

At this point you seem to be engaging in speculative and somewhat self-contradictory justifications for the absence of a definition, and taking me to task for asking the question. You're certainly not being helpful. If you doubt my question's importance, then I'm sure you have better things to do with your time.

In the meantime, I appreciate WWGD's understanding of the significance of the question. And that question is simple and factual: has anyone seen an explicit, stand-alone definition of "induces" in a mathematical context? Thanks.

9. Oct 9, 2015

Staff: Mentor

You seem to be agreeing with something I didn't say; i.e., that the usual uses of this word don't illuminate the mathematical meaning. In post #2 I gave one meaning that I found online, as well as several synonyms.
I am not arguing this. You cited two corpora in a previous post, neither of which gave a reasonable definition of "induce." A better choice would have been to consult just about any dictionary of English.
I have provided a definition, together with several synonyms. Whatever quibbles you have with this definition don't seem to me to have anything to do with mathematics.