[HN Gopher] The Two Ideals of Fields
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The Two Ideals of Fields
Author : susam
Score : 46 points
Date : 2025-05-31 13:59 UTC (9 hours ago)
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| mtsolitary wrote:
| This is a great example of one of those things in abstract maths
| that is hard to follow when you learn it, but once you've been
| through it a few times and learnt the definitions to heart it's
| really just a rephrasing of the definitions.
| tux3 wrote:
| Trivial in math is a term that refers to anything you've
| already learned.
|
| You sometimes hear people say that math is tautological. But
| regardless of whether it's all just an elaborate rephrasing of
| the axioms, it's quite beautiful.
| math_dandy wrote:
| Historically, mathematicians have spent a huge amount of time
| and effort formulating optimal axioms and foundations so that
| theorems would follow naturally from structure. Theorems
| following "trivially" from a theoretical framework that took
| years to develop isn't an indictment of the theorem, but an
| endorsement of the incredible effort expended to develop an
| optimal context for expressing and understanding the theorem.
| mathgradthrow wrote:
| Half of the work of mathematics is in correct definitions.
| Groethendieck referred to the division between mathematical
| labors as hunting and farming.
|
| This is not my most popular opinion, but probably the most
| consequential invention of the last 400 years was the set.
| Suddenly all mathematical knowledge could be verified in
| one framework. Physicists had a target in which to state
| their models.
|
| If you could state your hypothesis in the language of
| mathematics, "everyone" knew exactly what you meant by it,
| and how to go about testing your claims, or proving them,
| if they happened to be about mathematics itself.
|
| Calculus was invented in 1690ish, physicists like to claim
| that this was the most important advance in physics, but
| quantum mechanics and relativity didn't happen until
| dedekind invented the real numbers, 200 years later.
|
| It turns out that knowing what you're talking about
| matters.
| gjm11 wrote:
| Do you have any evidence that Dedekind's formalization of
| the real numbers was an essential step on the road to
| quantum physics and relativity? This seems very doubtful
| to me.
|
| A more plausible claim: the general move towards greater
| rigour in mathematics, one of whose expressions was
| Dedekind's formalization of the real numbers, improved
| the state of mathematical understanding in ways that were
| necessary for the arrival of quantum physics and
| relativity. E.g., to do quantum physics you want the
| notion of "vector space"; to do general relativity you
| want the notion of "Riemannian manifold"; to do special
| relativity maybe you want to have encountered the
| "Erlangen programme".
|
| But I'm not 100% convinced. It's not unusual for
| physicists to make use of mathematical notions that they
| don't have precise definitions of. E.g., I'm not sure
| anyone has an entirely satisfactory formal account of
| "path integrals"; string theory may or may not turn out
| to have anything to do with how the universe actually
| works, but if it doesn't it probably won't be because we
| don't have a complete account of what it _actually is_.
| Newton managed to do pretty impressive things with
| calculus before anyone had a really convincing definition
| of such advanced notions as, er, "derivative".
| mathgradthrow wrote:
| My evidence is just the timeline. Mathematics blew up
| right before physics did.
| aleph_minus_one wrote:
| > Trivial in math is a term that refers to anything you've
| already learned.
|
| According to a professor, "trivial" means: "If this is not
| trivial for you, you should see this as a clear signal that
| you should take this course seriously instead of slacking of,
| or even that you simply are in the wrong course."
| tekla wrote:
| This. It's always a good sign you've fucked up somewhere.
| kevinventullo wrote:
| I dunno, a common refrain I heard across all fields of math
| in grad school was "This is obvious. Wait, is this
| obvious...? Y... yes yeah it's obvious. "
| gosub100 wrote:
| My pet peeve math term is "clear". A long time ago I thought
| could teach myself group theory by buying the Springer group
| theory book and reading it from chapter 1, 1 page at a time.
| But I was blocked within the first 5 pages because the axioms
| and first few proofs kept saying how "clear" it was that all
| the results followed. Unfortunately, it was not "clear" to me
| :(
| JadeNB wrote:
| > the Springer group theory book
|
| I am skeptical that this uniquely identifies a book (unless
| you mean the book "Linear Algebraic Groups" by the author
| called Springer, rather than the publisher called Springer,
| in which case it's definitely not the way to start learning
| group theory!).
| gosub100 wrote:
| It was the yellow Springer publishing book. Happened 20+
| years ago now, cannot recall the author. IIRC the title
| was "a course in the theory of groups".
| layer8 wrote:
| This one, probably:
| https://link.springer.com/book/10.1007/978-1-4419-8594-1
|
| It's a graduate-level text, to be fair.
| mwcremer wrote:
| I had a calc prof who was in the middle of a lecture,
| "...and as any fool can see, X is..." He stopped, turned
| around, and said, "You know, sometimes when I say, 'It is
| intuitively obvious', or, 'As any fool can see', I realize
| it may _not_ be intuitively obvious, and any fool may _not_
| be able to see. But as any fool can see, X is... "
| downboots wrote:
| How to distinguish mastery of a complex subject from parsing
| one formally expressed in a complex way?
| aleph_minus_one wrote:
| > How to distinguish mastery of a complex subject from
| parsing one formally expressed in a complex way?
|
| In my opinion: the difference between a complex subject and
| one formally expressed in a complex way is that in the
| former, the results that you get are really deep
| (understanding them at the end feels like a spiritual
| experience).
| mtsolitary wrote:
| I think what you call "parsing" is largely indistinguishable
| from mastery in a lot of fields, particularly in abstract
| mathematics
| fn-mote wrote:
| I disagree. "Parsing" is the first level of understanding.
| If you are not moving past the parsing level, you have not
| achieved any kind of mastery.
|
| My experience is that mastery means more like "you have a
| mental model which gives you 'intuitive' reasons to
| accurately classify things as true/false and provides some
| motivation for the reasoning".
|
| An example: you see someone has solved a degree 4 equation
| by repeatedly applying the quadratic equation, getting 8
| solutions. "No way."
|
| Another example: watch a famous baking show and you see
| somebody put a bunch of different sized pieces of bread in
| the oven at the same time. Right away: "aren't they going
| to cook at different rates?" Sure enough, some burned, some
| raw.
| layer8 wrote:
| Knowing the rules of chess doesn't make you a chess master.
| Knowing the syntax and semantics of a programming language
| doesn't make you a master software architect.
| bubblyworld wrote:
| Yeah, as an example this is a simple corollary of the following
| result which gets used a lot: the quotient of a (unital,
| commutative) ring by a maximal ideal is always a field.
|
| If your ring has only two ideals then the trivial ideal is
| maximal, and thus your ring is already a field!
|
| The more you know, the more "shortcuts" you start seeing, I
| guess.
| srean wrote:
| Spivak states in his "Calculus on Manifolds' that definitions
| should be hard (to refine and state) and when done well, the
| theorems easy.
| almostgotcaught wrote:
| this is a standard thing in "mature" areas of math and it's
| absolutely the opposite of what's good for the student (all
| of the machinery being hidden in the definition instead of
| developed in the theorem's proof).
|
| EDIT: if you hate "a monad is a monoid in the category of
| endofunctors" then you also hate "definitions should be hard
| and theorems easy".
| andrewflnr wrote:
| My first instinct is to agree, but I'm not sure actually.
| What I really want when learning a new area of math is the
| full motivation for the tricky definition, taking as much
| time as needed to follow the dead ends of easier but worse
| definitions. Then I get the whole picture. IMO the
| motivation is the key thing for students, not the
| definition being easy.
|
| Though maybe the way this course would work is in fact by
| proceeding through a series of easy but explicitly flawed
| definitions, and proving both real results and nonsense
| from them, so you see why the real definition is justified.
| almostgotcaught wrote:
| > What I really want when learning a new area of math is
| the full motivation for the tricky definition
|
| yes so then you want proofs that actually exercise real
| machinery instead of playing the shell game of "an X is a
| Y and a Y is a Z, and has ABC properties, there for X has
| ABC properties"; you want a proof that goes through the
| process of using properties ABC to build Y from Z and X
| from Y (or something akin to that).
|
| definitions aren't for people learning math, they're for
| people using math ie practising professional
| mathematicians that are proving more theorems; Hausdorff
| didn't invent "Hausdorff spaces", he used/worked with
| various properties of topological spaces and then when
| the next person came along and needed to right another
| paper on top, that person invented "Hausdorff space".
| kevinventullo wrote:
| FYI many sources do not count the entire ring as an ideal. If you
| do, you'd have to define "maximal ideal" to mean "an ideal that
| is maximal with respect to inclusion, _ignoring_ the entire ring
| ideal."
| vouaobrasil wrote:
| Pretty much most sources, actually. When actually working with
| ideals, there's almost never an advantage to consider the whole
| ring an ideal. So all ideals in virtually all the literature
| I've ever read were taken to be proper, i.e. proper subsets
| WCSTombs wrote:
| I very strongly disagree. Without considering the whole ring
| to be an ideal, you can't even define "the ideal generated by
| some elements" because there may not be such an ideal, since
| it could be the whole ring. Likewise, you can't perform
| common operations on ideals like sums because they could
| result in the whole ring. In fact, I would say _there is
| almost no advantage in excluding the ring itself from the set
| of ideals_.
|
| Just to check, I have three math textbooks from my college
| days that include the definition of an ideal, and none of
| them attempt to exclude the ring itself from the definition.
|
| The obvious compromise is to introduce the concept of a
| _proper ideal_ as an ideal that is a proper subset, and to
| use that when you need to exclude the ring itself. E.g., a
| maximal ideal is a proper ideal that is maximal with respect
| to inclusion.
| kevinventullo wrote:
| Too late to edit, but you're right I was misremembering. I
| was thinking of whether (1) should be considered a prime
| ideal.
| Sniffnoy wrote:
| That's not my experience at all as a mathematician. "Maximal
| ideal" implicitly means "maximal proper ideal", yes, but
| generally ideals don't have to be proper unless specified so.
|
| If you don't include the whole ring as an ideal, you can't
| even define ideal addition, etc. I took an algebra class once
| from a professor who decided to define "ideal" to mean
| "proper ideal". After a few weeks he had to give it up
| because it just became too much trouble for reasons like
| that; he had to too often say "possibly improper ideal",
| i.e., this convention had the opposite effect he intended! I
| can't think of any other source I've seen use that
| convention.
| vouaobrasil wrote:
| Another interesting fact about fields is that a commutative ring
| is a field if and only if every ideal is a prime ideal.
| (Obviously, every ideal of a field is a prime ideal. The converse
| is more interesting...)
| shiandow wrote:
| This is one of those cases where the algebraic geometry
| perspective is actually helpful (without too much trouble).
|
| If the only subspaces are single points then the space is
| itself a point.
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