What is the relationship between the concept of field in physics and math?

[cesiumfrog]

"field" in physics vs math

The word "field" seems to describe very different notions in physics (a function that maps from some space to .. another) versus mathematics (a set having addition and multiplication). Is there any conceptual or historic link to explain why the name is used for both notions?

 

[Nolen Ryba]

It has nothing to do with physics vs. math. The two different entities of a vector (or tensor) field and an algebraic field were assigned the same name. Both entities occur in physics and math.

 

[cesiumfrog]

OK, but why were both entities assigned the same name? Do you know what motivated either assignment?

 

[Math Jeans]

OK, but why were both entities assigned the same name? Do you know what motivated either assignment?

Because they both depict a group of vectors over a space.

 

[HallsofIvy]

?? A mathematical field certainly does NOT "depict a group of vectors over a space".

 

[Math Jeans]

Then what do you consider to be a differential graph?

 

[morphism]

And what exactly is a differential graph? Google shows no real definition, and certainly nothing related to fields (in the algebraic sense of the word).

(Irrelevant side note: We can think of a field as a "group of vectors over a space", more precisely as a vector space over its prime subfield. This is irrelevant because I think the word "field" was used because it was the English [or maybe more accurately German? Maybe not!] word that the person who invented the object thought would be fitting.)

 

[Math Jeans]

a differential or "vector" graph creates a vector field in which a group of vectors flow in the general direction of the intended graph.

Basically a vector field.

 

[Nolen Ryba]

OK, but why were both entities assigned the same name? Do you know what motivated either assignment?

Replies to questions like this always run the risk of going into some sort of pseudophilosophical debate, but what the hell, sometimes they provide some insight. The vector/tensor field seems natural enough... "field" typically refers to something spatial like that... or you could say a vector field looks like a grassy field blowing in the wind. The algebraic field, well, I don't know, but I also don't understand the motivation for the following algebraic terms:

-magma (wtf)
-group
-ring
-domain (the integral I get ;))
-module

Perhaps these entities are just so far removed from every day experience that no name which gives you a rough idea of what they are could be assigned? Perhaps I'm just ignorant.

 

[HallsofIvy]

a differential or "vector" graph creates a vector field in which a group of vectors flow in the general direction of the intended graph.

Basically a vector field.

Thank you. Now back to my point. The field of real numbers certainly does not depict a group of vectors over a space". Neither does the field or rational numbers nor the field of complex numbers. And certainly a finite field doesn't have anything to do with vectors! What does that have to do with a "differential graph"?

 

[Chris Hillman]

?? A mathematical field certainly does NOT "depict a group of vectors over a space".

Not sure what you mean here, Halls, but the algebraic notion of field originates with the famous memoir of Galois, where indeed a splitting extension field of the field of rationals is among other things a vector space over the base field.

The algebraic field, well, I don't know, but I also don't understand the motivation for the following algebraic terms:

• magma (wtf)
• group
• ring
• domain (the integral I get ;))
• module

These concepts were all introduced in papers written in French or German, but I'll try to give some motivation for the English terms:

• group: Galois was discussing what we'd now call cosets (the French term used by Bourbaki for the left coset $g \, H$ is classe a gauche modulo H; German Nebenkomplex), "groups" of permutations which "behave the same" under composition with a given permutation, thus forming a quotient group in favorable cases. The word was invented much later: IIRC, by Dedekind (German Gruppe, French groupe).
• ring: think of $Z/\left( n \, Z \right)$ as a "circle". IIRC, Hilbert introduced the German term, ZahlRing ("number ring"). The French word is anneau ("ring" or "band"); see http://mathworld.wolfram.com/Ring.html
• module: the concept and the German term Zahlenmodul or simply Modul are due to Dedekind (1871); see http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Ring_theory.html. The modern French word is module, which probably explains the origin of the English word.
• domain: the concept arose from Dedekind's work on "factorizing" ideals; he referred to a "domain of rationality" for what we'd call an "integral domain". I presume the English word was chosen as a word with similar connotations to "field" which could not however be easily confused with "field". BTW, the French term used by Bourbaki is anneau d'integrite. The modern German term is Integraetsbereich. The German term for "domain" of a mapping is Bereich or Definitionsmenge.
• field: the concept first appears in the work of Abel and Galois; the German term Koerper (lit. "corpus" or "body") was introduced by Dedekind and later translated into English as "field". (I once knew but have forgotten who did that.) The French term is corps. The German term for a "field" of mathematics is Disziplin, or sometimes Gebiet.
• magma: the concept (a set equipped with a well-defined binary operation with no other assumptions) and the French term was introduced by Bourbaki to suggest "minimal structure"; see webpages which are authoritative or not, as you please.

No doubt mathematically literate native speakers of German or French will correct any inadvertent errors I might have made concerning mathematical terminology in those languages!

The now standard definitions of the terms learned by modern students were generally introduced around 1920 in German texts (e.g. by Van der Waerden) or French ones (e.g. by Bourbaki). Earlier writers were quite sloppy by modern standards, which can sometimes be a real obstacle in reading papers written before 1920.

I have not seen the book by Steven Schwartzman, The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English, Spectrum, but it was well reviewed by Henry J. Ricardo in Am. Math. Monthly 102 (1995):563-565, so this should be just what you want!

 

[Nolen Ryba]

Wow, I appreciate the sources Chris! The historical development of the terminology (which in my experience often includes insights) is not something that is commonly taught. I'll keep that information in mind for (1) my own studies and (2) when I teach the subject.

 

[cesiumfrog]

field: the concept first appears in the work of Abel and Galois; the German term Koerper (lit. "corpus" or "body") was introduced by Dedekind and later translated into English as "field". (I once knew but have forgotten who did that.) The French term is corps. The German term for a "field" of mathematics is Disziplin, or sometimes Gebiet.

It seems the "translation" was by Eliakim Hastings Moore, though I'd be curious for any rationalisation of why he found the word "field" (rather than using something with a similar meaning, or even sound, to the German original).

Would he have thought the Galois-"field" connected it somehow to vector "feld"s?

 

[Civilized]

Once I was browsing the math libraries listings on algebraic fields, when I came across Atiyah's classic "Geometry of Yang-Mill's fields", which is a book about the kind of quantum fields used in the standard model of particle physics, and for a moment I thought it might hold the key to understanding how these structures are related, but of course it had just been poorly placed by a librarian.