[HN Gopher] Galois Theory
___________________________________________________________________
Galois Theory
Author : mathgenius
Score : 441 points
Date : 2024-08-15 13:00 UTC (1 days ago)
(HTM) web link (golem.ph.utexas.edu)
(TXT) w3m dump (golem.ph.utexas.edu)
| 082349872349872 wrote:
| In particular, abuse of Galois Theory makes it possible to
| reconcile Spinoza with Aquinas.
| esafak wrote:
| Go on.
| koolala wrote:
| Reconcile Liebniz too. I agree this sounds awesome.
| 082349872349872 wrote:
| Leibniz _does_ get a mention, but it 'll already take a
| while to explain these two, so I'll leave his full
| reconciliation as an exercise for the reader ;)
| koolala wrote:
| ;) A mention is enough. Monads are LIFE!
| 082349872349872 wrote:
| Every mathematical monad comes from an adjunction, which
| unfortunately implies they're not as windowless as
| Leibniz would like? (or does it merely imply that in the
| Leibnizian setting the structure is always external and
| never internal?)
|
| Can we make a mathematical monad out of the adjunction
| presented above? Let g(C) be the minimum G and c(G) be
| the maximum C in the model above, then we certainly have
| g(C) = g(c(g(C))) and c(G) = c(g(c(G))), which formally
| suggest we may be able to do something.
|
| Exercise: does the triple (T, m, e) exist? what about (G,
| d, [?]) in the other direction? If they do exist, what
| are they, for Aquinas, Leibniz, and Spinoza?
| novosel wrote:
| Why would they need to be reconciled? Or, indeed, Galoas
| abused?
| 082349872349872 wrote:
| They don't _need_ to be reconciled, but they do differ, which
| suggests we might be able to find a framework which
| reconciles them.
|
| _Galois_ _Spinoza_ _Aquinas_ in Google yields (for me at
| least) the following HN thread:
| https://news.ycombinator.com/item?id=39885475 , in which an
| attempt at "Algebraic Theology" not only provides such a
| subsuming (thanks to Galois Theory) framework, answering that
| particular question in the affirmative, but also raises many
| other questions which might be amusing to pursue. (I could
| summarise those Q's in this thread, if any of you all are
| more interested than the downvoters were)
|
| [I will summarise that thread further up in this one, but as
| it was months ago it may take me an hour or two to page
| everything back in]
| 082349872349872 wrote:
| To expand[0] upon that claim:
|
| Aquinas, Leibniz, and Spinoza all agree that there is a God (G)
| that created[1] a Creation (C) which we are a part of.
|
| One major way Aquinas (and Leibniz) differ from Spinoza is in
| how determinate C may be. Spinoza says C is determined by G;
| Aquinas says there are many possible C's for any given G.
| (Leibniz splits the difference and says there are many possible
| C's, but in our particular case, our G has created the best[2]
| possible C.)
|
| The reconciliation: let G and C be in an adjoint relationship,
| such that we have functions picking out the maximum C any given
| G may create, and the minimum G that can create any given C.
|
| Now, if you are Aquinas, G is omnipotent, and hence has the
| possibility to create other C's, but our C is the maximum[3]
| one.
|
| On the other hand, if you are Spinoza, G determines C[4], so it
| is trivially maximal. (The maximum of a singleton being the
| unique element)
|
| Does that make sense?
|
| Question: do there exist Gods that are incapable of creating
| any creation, or Creations that are impossible for any god to
| create? If so, need we replace "maximum" and "minimum" above by
| LUB and GLB? What other situations (eg. gods or creations being
| only domains rather than lattices) would also require further
| abstraction?
|
| :: :: ::
|
| [0] my apologies for any non-standard notation. I tried to find
| a survey paper on "Algebraic Theology" so I could follow the
| existing notation, but failed to find any concrete instances in
| this (currently only platonic?) field.
|
| [1] Spinoza is accused of "pantheism": the heresy of
| identifying God and the Creation. Reading him according to a
| Galois-theoretic model, he would be innocent of this heresy,
| for when he says "God, or, the Universe", he is simply using
| metonymy, for in his model God and Creation are dual, so (being
| in a 1:1 relationship) one determines the other. Note that in
| general, not only are duals not identical, they're not even
| isomorphic.
|
| [2] but cf Voltaire, _Candide_ (1759)
|
| [3] if you are Leibniz, the order in which it is maximal
| corresponds to the traditional "worst" "better" "best" order. I
| don't know Leibniz well enough to say if he had a total, or
| only partial, order in mind; presumably in his model if there
| are several maximal "best" creations they would all be
| isomorphic?
|
| Exercise for the reader: work out the Leibnizian metaphysical
| adjunction.
|
| [4] turning this arrow around, C determines G, which explains
| why Einstein would say he believed in the "God of Spinoza", and
| chose to base his research by thinking about C, unlike the
| medieval colleagues of Aquinas, who spent a lot of time and
| effort trying to work the arrow in the other direction, hoping
| to come to conclusions about C in starting by thinking about G.
| gowld wrote:
| How does Galois theory relate to any of this?
| rpmw wrote:
| I will always remember Galois theory as the punchline to my
| Abstract Algebra courses in college. Galois was a brilliant math
| mind, and I'm curious what else he would have contributed had he
| not died at 20 in a duel.
|
| https://en.wikipedia.org/wiki/%C3%89variste_Galois
| mcfunley wrote:
| Great to see this course material public. It's a real missed
| opportunity though to not mention that Galois wrote a lot of it
| down staying up all night before being shot.
| bubble12345 wrote:
| That's a common myth. See this paper referenced in the
| wikipedia article:
|
| Rothman, Tony (1982). "Genius and Biographers: The
| Fictionalization of Evariste Galois". The American
| Mathematical Monthly. 89 (2): 84-106. doi:10.2307/2320923.
| JSTOR 2320923
| ForOldHack wrote:
| Quite the clickbait. You can only access it from the pay
| site, or unless you can get a school library to access it,
| which I will do. Only the first page is available free.
| bubble12345 wrote:
| Or you can paste the following JSTOR link into sci-hub.
|
| https://www.jstor.org/stable/2320923
| agumonkey wrote:
| I'd be genuinely curious to have a chat with similar minded
| people. They must see the world quite differently to be able to
| fork a new path in hard maths mostly on their own ..
| frakt0x90 wrote:
| My second semester of algebra had a section on Galois theory and
| I remember thinking it was abstract nonsense and I didn't get it.
| I'm actually interested in going through this to see if my
| perspective has changed.
| openasocket wrote:
| It's all abstract nonsense until it starts having practical
| results lol :)
|
| The main motivation for Galois theory was proving the
| insolvability of the quintic. For those not aware, there is a
| general formula solving the quadratic equation (i.e. solving
| ax^2 + bx + c = 0). That formula has been known for millenia.
| With effort, mathematicians found a formula for the cubic (i.e.
| solving ax^3 + bx^2 + cx + d = 0), and even the quartic (order
| 4 polynomials). But no one was able to come up with a closed-
| form solution to the quintic. Galois and Abel eventually proved
| that a quntic formula DOES NOT EXIST. At least, it cannot be
| expressed in terms of addition, subtraction, multiplication,
| division, exponents and roots. You can even identify specific
| equations that have roots that cannot be expressed in those
| forms, for example x^5 - x + 1.
|
| I took an entire course on it that went through the proof. It's
| actually very interesting. It sets up this deep correspondence
| between groups and fields, to the point that any theory about
| groups can be translated into a theory about fields and vice
| versa. And it provides this extremely powerful set of tools for
| analyzing symmetries. The actual proof is actually really
| anticlimactic. You have all these deep proofs about the
| structures of roots of polynomial equations, and at the very
| end you just see that the structures of symmetries of certain
| polynomials of degree 5 (like x^5 - x + 1) don't follow the
| same symmetries that the elementary mathematical operations
| have. Literally, the field of solutions to that polynomial
| doesn't map to a solvable group:
| https://en.wikipedia.org/wiki/Solvable_group
|
| In almost the same breadth, it also proves that it is
| impossible to trisect an angle using just a compass and
| straightedge, a problem that had been puzzling mathematicians
| for millennia. It's actually almost disappointing: we spent
| then entire course just defining groups and fields and field
| extensions and all this other "abstract nonsense". And once all
| of those definitions are out of the way, the proof of the
| insolvability of the quintic takes 10 minutes, same for the
| proof of impossibility of trisecting an angle.
| koolala wrote:
| Did the solution taking 10 minutes make it seem like it was
| all just semantics from old faulty definitions?
|
| How do you personally imagine trisecting an angle now? Is it
| possible to describe your new intuition of the impossibility
| in different human understandable terms that are also
| geometric? Impossible things are weird conversation subjects.
| openasocket wrote:
| > Did the solution taking 10 minutes make it seem like it
| was all just semantics from old faulty definitions?
|
| I recall that I, and the rest of the class, were very
| suspicious of the proof. The proof took maybe 10 minutes,
| but it probably took another 10-15 minutes for the
| professor to convince us there wasn't a logical error in
| the given proof. Though the situation was kind of the
| opposite of what you would thing. We understood field
| extensions and the symmetries of the roots of polynomials
| really well. What took convincing was that any formula
| using addition, subtraction, multiplication, division,
| exponents, and rational roots would always give you a field
| extension that mapped to a "solvable group". The proof is
| essentially:
|
| 1. Any field extension of a number constructed using those
| mathematical operations must map to a solvable group. 2.
| For every group there exists a corresponding field
| extension (this is a consequence of the fundamental theorem
| of Galois theory). 3. There exist groups that are not
| solvable. 4. Therefore, there are polynomials with roots
| that can't be constructed from the elementary mathematical
| operations.
|
| Basically the entire course is dedicated to laying out part
| 3, and the part we were suspicious about was part 1.
|
| The one thing that is interesting about the proof is that
| it is actually partially constructive. Because there is no
| general quintic formula, but there are some quintics that
| are solvable. For instance, x^5 - 1 clearly has root x=1.
| And Galois theory allows you to tell the difference between
| those that are solvable and those that are not. It allows
| you to take any polynomial and calculate the group of
| symmetries of those roots. If that group is solvable, then
| all of the roots can be defined in terms of elementary
| operations. If not, at least one of the roots cannot.
|
| > How do you personally imagine trisecting an angle now? Is
| it possible to describe your new intuition of the
| impossibility in different human understandable terms that
| are also geometric?
|
| So the trisection proof I don't remember as well, but
| looking it up it isn't very geometric. It essentially
| proves that trisecting an angle with a compass and straight
| edge is equivalent to solving certain polynomial equations
| with certain operations, and goes into algebra.
|
| That said, Galois theory itself feels very "geometric" in
| the roughest sense of the term. Fundamentally, it's about
| classifying the symmetries of an object.
| kevinventullo wrote:
| _2. For every group there exists a corresponding field
| extension (this is a consequence of the fundamental
| theorem of Galois theory)._
|
| Just a nit, but when talking about extensions of Q, this
| is called the Inverse Galois Problem and it is still an
| open problem.
|
| That said, you don't actually need this strong of a
| statement to show general insolvability of the quintic.
| Rather you just need to exhibit a _single_ extension of Q
| with non-solvable Galois group. I believe adjoining the
| roots of something like x^5+x+2 suffices.
| koolala wrote:
| Your story makes me picture Geometric Algebra, defining
| Complex Numbers or Quaternions and multiplication on
| them, and their symmetries and elegant combinations. Ty
| for sharing your math memories. Makes me wonder if Galois
| theory can determine valid or invalid imaginary number
| combinations / systems.
| gowld wrote:
| The reason that the proof isn't geometric, is that the
| algebriac proof is a proof that Euclidean geometry is
| incomplete. How can you use a language (any given
| language!) to express the idea that the selfsame language
| is incapable of expressing a certain concept?
|
| You can draw a picture of trisecting an angle using an
| ruler (with cube-root markings) or an Archimedian sprial,
| which are clearly more powerful than purely Euclidean
| geometery, but how can you draw a picture of it being
| _impossible_ without something like this?
|
| How do you draw a picture of something that doesn't
| exist?
|
| You can draw pictures of what does exist, like the
| symmetries in Arnold's proof of unsolvability of the
| quintic https://mcl.math.uic.edu/mcl.math.uic.edu/wp-
| content/uploads... and show that those symmetries can do
| things that radicals can't.
|
| I don't know of a similar visual for non-trisectability
| of angles.
| ForOldHack wrote:
| "the proof of impossibility of trisecting an angle."
|
| It makes it into a silly little footnote, a very little
| footnote, I was both sad and disappointed when I read and
| understood it in an appendix.
|
| Now back to Pi + e = Pie.
| poincaredisk wrote:
| It solves a problem that was puzzlimg mathematicians for
| millennia - I wouldn't call that a footnote. You may not
| consider pure math applications interesting, but that
| doesn't make them unimportant.
|
| Galois theory is big today because it provides a connection
| between a ring theory and field theory. This has huge
| applications for other branches of modern math - for
| example, number theory. But that always culminates in a
| pure math application, which makes it a footnote, I guess.
| agumonkey wrote:
| Perspective change are wonderful. Sometimes you realize late in
| life that people from 100+ years ago were so advanced
| intellectually.
| 082349872349872 wrote:
| On the other hand, getting yourself shot in the gut 152 years
| before the invention of the Bogota bag might be a textbook
| example of the english chengyu "book smart life stupid"?
|
| (see
| https://vuir.vu.edu.au/18204/1/PETSINIS_1995compressed.pdf
| but note that as well as being a fictional account, Petsinis
| has made some factual errors in details taken from the
| coroner's report:
| https://news.ycombinator.com/item?id=40650555 )
| gradschoolfail wrote:
| I hear you brother! For me, people from 1 month ago!
| VyseofArcadia wrote:
| Chapter 1 is brilliant.
|
| I've been shouting from the rooftops for years that math[0]
| courses need more _context_. We can prove X, Y, and Z, and this
| class will teach you that, but the motivating problem that led to
| our ability to do X, Y, and Z is mentioned only in passing.
|
| We can work something out, and then come back and rework it in
| more generality, but then that reworking becomes a thing in and
| of itself. And this is great! Further advances come from doing
| just this. But for pedagogical purposes, stuff sticks in the
| human brain so much better if we teach the journey, and not just
| the destination. I found teaching Calculus I was able to draw in
| students so much more if I worked in what problems Newton was
| trying to solve and why. It gave them a story to follow, a reason
| to learn this stuff.
|
| Kudos to the author for chapter 1 (and probably the rest, but
| chapter 1 is all I've had time to skim).
|
| [0] And honestly, nearly every subject.
| gowld wrote:
| Hot take:
|
| "18th Century" mathematics was intiuitive and informal, to the
| point that it was inconsistent.
|
| The 19th and 20th Centuries added rigor and formalism (and
| elitism) and _devalued intuition_ , to the point that it begame
| uninterpretable to most.
|
| The 21st Century's major contribution to mathematics (including
| YouTube! and conversational style writing) was to bring back
| intuition, with the backing of formal foundations.
| Agingcoder wrote:
| I like your idea.
|
| I was schooled in abstract 20th century math - indeed YouTube
| is the opposite, and it's a good thing.
|
| One of my math teachers was once talking to Jean Dieudonne
| https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9
|
| who was part of the Bourbaki group and asked him why on earth
| he insisted on inflicting raw dry theory to the world with no
| intuition , when his day job involved drawing ideas all day
| long !
|
| Edit: interestingly enough, one of my colleagues thinks very
| strongly that intuition should not be shared, and the path to
| intuition should be walked by everyone so that they ' Make
| their own mental images '. I guess that there's a tradeoff
| between making things accessible, and deeply understood, but
| I don't know what to make of his opinion.
| wging wrote:
| Do you know how Dieudonne answered?
| Koshkin wrote:
| _On foundations we believe in the reality of mathematics,
| but of course, when philosophers attack us with their
| paradoxes, we rush to hide behind formalism and say
| 'mathematics is just a combination of meaningless
| symbols,'... Finally we are left in peace to go back to
| our mathematics and do it as we have always done, with
| the feeling each mathematician has that he is working
| with something real. The sensation is probably an
| illusion, but it is very convenient._
| stoneman24 wrote:
| The 'make thier own mental models' vs sharing/providing
| full information is difficult.
|
| 'Make thier own model' of the domain can lead to deeper
| understanding but takes time and may lead to different
| (possibly incorrect) understanding of the issues and
| complexities. If not reviewed with others.
|
| Providing full information upfront to a person can be
| quicker but lead to a superficial knowledge.
|
| I think that it comes down to whether that deeper knowledge
| is directly needed for the main task. Can I get by with an
| superficial (leaky) abstraction and concentrate on the main
| job.
| fredilo wrote:
| > interestingly enough, one of my colleagues thinks very
| strongly that intuition should not be shared, and the path
| to intuition should be walked by everyone so that they '
| Make their own mental images '. I guess that there's a
| tradeoff between making things accessible, and deeply
| understood, but I don't know what to make of his opinion.
|
| If the objective is to advance mathematics instead of
| making it accessible, then this is a somewhat reasonable
| position. The mathematical statements that a person can
| come up with is often a direct product of their mental
| image. If everyone has the same image, everyone comes up
| with similar mathematical statements. For this reason you
| want to avoid that everyone has the same picture. Forcing
| everyone to start with a clean canvas increases the chance
| that there is diversity in the images. Maybe someone finds
| a new image, that leads to new mathematical statements. At
| least that's the idea. One could also argue that it just
| leads to blank canvases everywhere.
| paulddraper wrote:
| Elements is formal, right?
| hollerith wrote:
| Pretty much. According to someone on Stack Exchange, "The
| most serious difficulties with Euclid from the modern point
| of view is that he did not realize that an axiom was needed
| for congruence of triangles, Euclids proof by superposition
| is not considered as a valid proof." But making mistakes in
| a formal treatment of a subject does not negate the fact
| that it is a formal treatment of the subject IMHO, and
| AFAICR none of the theorems in Elements is wrong; i.e.,
| Elements is formal enough to have avoided a mistake even
| though the axioms listed weren't all the axiom that are
| actually needed to support the theorems.
|
| 18th Century European math was much more potent than
| ancient Greek math, and although parts of it like algebra
| and geometry were, for a long time, most of it was not
| understood at a formal or rigorous level for a long time
| even if we accept the level of rigor found in Elements.
| koolala wrote:
| Isn't how formal or rigorous something is just a social
| convention? Grammer Nazi's used to make online speech be
| formal with perfect rigor. Isn't it all relative to what
| your society defines?
| chongli wrote:
| No. That's the colloquial definition of formal. In
| mathematics, the word formal refers to something more
| specific: one or more statements written using a set of
| symbols which have fully-defined rules for mechanically
| transforming them into another form.
|
| A formal proof is then one which proceeds by a series of
| these mechanical steps beginning with one or more
| premises and ending with a conclusion (or goal).
| koolala wrote:
| Both are a formality based on whats in fashion. I like
| the Axiom of Choice and not taking Math or words as
| literal or biblical truth.
| chongli wrote:
| If you're a formalist in philosophy of math, then math is
| neither true nor false, it's merely a bunch of
| meaningless symbols you transform via mechanical rules.
| nyssos wrote:
| To an extent. A truly completely _formal_ proof, as in
| symbol manipulation according to the rules of some formal
| system, no. It 's valid or it isn't.
|
| But no one actually works like this. There are varying
| degrees of "semiformality" and what is and isn't
| acceptable is ultimately a convention, and varies between
| subfields - but even the laxest mathematicians are still
| about as careful as the most rigorous physicists.
| gowld wrote:
| Elements is mostly formal, but it's also concrete and
| visual.
|
| Euclid developed arithmetic and algebra through
| constructive geometry, which relies on our visual intuition
| to solve problems. Non-concrete problems were totally out
| of scope. Even curved surfaces (denying the parallel
| postulate) were byond Euclid. Notably, Elements didn't have
| imaginary or transcendental numbers. Euclid made no attempt
| to unify line lengths and arc lengths, and had nothing to
| say about what fills the gaps between the algebraically
| (geometrically!) constructible numbers.
| paulddraper wrote:
| > Elements didn't have imaginary or transcendental
| numbers
|
| Elements has pi.*
|
| *It proves the ratio of a circle's area to the square of
| its diameter is constant.
| gowld wrote:
| But doesn't say how large that constant value is.
| Someone wrote:
| > Euclid [...] had nothing to say about what fills the
| gaps between the algebraically (geometrically!)
| constructible numbers
|
| Did he even know those gaps existed? Euclid lived around
| 300BC. The problem of squaring the circle had been
| proposed around two centuries before that
| (https://en.wikipedia.org/wiki/Anaxagoras#Mathematics),
| but I don't think people even considered it to be
| impossible by that time.
| Koshkin wrote:
| Incidentally, here's one beautiful edition:
|
| https://farside.ph.utexas.edu/Books/Euclid/Elements.pdf
| dataflow wrote:
| Do they explain anywhere why we consider radicals special and
| ask these questions about them in the first place? Nobody ever
| explains that. To my programmer brain, whether I'm solving t^3
| = 2 or t^100 + t + 1 = 7 numerically, I have to use an
| iterative method like Newton's either way. ("But there's a
| button for Nth root" isn't an argument here - they could've
| added a more generic button for Newton.) They don't
| fundamentally seem any different. Why do I care whether they're
| solvable in terms of _radicals_? It almost feels as arbitrary
| as asking whether something is solvable without using the digit
| 2. I would 've thought I'd be more interested in whether
| they're solvable in some other respect (via iteration, via
| Newton, with quadratic convergence, or whatever)?
| Joker_vD wrote:
| I imagine it has something to do with the fact that there is
| a somewhat simple pen-and-paper procedure, quite similar to
| long division, for calculating square/cubic/etc. roots to
| arbitrary precision, digit by digit. So finding roots is, in
| a sense, an arithmetic operation; and of calculus one better
| not speak in a polite society.
| phs wrote:
| Radicals are a "natural" extension in a certain sense, just
| as subtraction and division are. They invert an algebraic
| operation we often encounter when trying to solve equations
| handed to us by e.g. physics. I find it understandable to
| want to give them a name.
|
| Why not something like "those things we can solve with
| Newton"? As you note Newton is broadly applicable; one would
| hope, given how popular the need to invert an exponent is,
| that something better (faster, more stable) if more specific
| than Newton might be created. It is hard to study a desired
| hypothetical operation without giving it a name.
|
| On a related note, how come we don't all already have the
| names of the 4th order iterative operation (iterated
| exponents) and its inverse in our heads? Don't they deserve
| consideration? Perhaps, but nature doesn't seem to hand us
| instances of those operations very often. We seemingly don't
| need them to build a bridge or solve some other common
| practical engineering problem. I imagine that is why they
| fail to appear in high school algebra courses.
| bubble12345 wrote:
| You are right in the sense that solvability by radicals has
| no practical importance, especially when it comes to
| calculations.
|
| It is just a very classical pure math question, dating back
| hundreds of years ago. Its solution led to the development of
| group theory and Galois theory.
|
| Group theory and Galois theory then are foundational in all
| kinds of areas.
|
| Anyway, so why care about solvability by radicals? To me the
| only real reason is that it's an interesting and a natural
| question in mathematics. Is there a general formula to solve
| polynomials, like the quadratic formula? The answer is no -
| why? When can we solve a polynomial in radicals and how?
|
| And so on. If you like pure math, you might find solvability
| by radicals interesting. It's also a good starting point and
| motivation for learning Galois theory.
| VyseofArcadia wrote:
| > To my programmer brain, whether I'm solving t^3 = 2 or
| t^100 + t + 1 = 7 numerically
|
| Typically when you're solving a polynomial equation in an
| applied context (programming, engineering, physics,
| whatever), it's because you have modeled a situation as a
| polynomial, and the information you really want is squirreled
| away as the roots of that polynomial. You don't actually care
| about the exact answer. You've only got k bits of precision
| anyway, so Newton's method is fine.
|
| But we're not interested in the solution. We don't
| particularly care that t = approx. 1.016 is a numerical
| solution.[0] We're not using polynomials to model a
| situation. In mathematics, we are often studying polynomials
| as objects in and of themselves, in which case the kind of
| roots we get tells us something about polynomials work, or we
| are using polynomials as a lens through which to study
| something else. In either case it's less about a specific
| solution, and more about what kind of solution it is and how
| we got it.
|
| Not to mention, specific polynomials are examples. Instead of
| t^100 + t + 1 = 7, we're usually looking at something more
| abstract like at^100 + bt + c.
|
| [0] And in the rare case we actually care about a specific
| root of a specific polynomial, and approximate numerical
| solution is often not good enough.
| photonthug wrote:
| > Do they explain anywhere why we consider radicals special
| and ask these questions about them in the first place? Nobody
| ever explains that. To my programmer brain,
|
| But regardless of whether you're a programmer, mathematician,
| machinist, carpenter, or just a kid playing with legos,
| there's _always_ a good time to be had in the following way:
| first you look at the most complex problems that you can
| manage to solve with simple tools; then you ask if your
| simple tools are indeed the _simplest_ ; and then if multiple
| roughly equivalently simple things are looking tied in this
| game you've invented then now you get the joy of endlessly
| arguing about what is most "natural" or "beautiful" or what
| "simple" even means _really_. Even when this game seems
| pretty dumb and arbitrary, you 're probably learning a lot,
| because even when you can't yet define what you mean by
| "simple" or "natural" or "pretty" it's often still a useful
| search heuristic.
|
| What can you do with a lot of time and a compass and a ruler?
| Yes but do we need the "rule" part or only the straight-edge?
| What can we make with only SKI combinators? Yes but how
| awkward, I rather prefer BCKW. Who's up for a round of code-
| golf with weird tiny esolangs? Can we make a zero
| instruction-set computer? What's the smallest number of tools
| required to put an engine together? Yes but is it
| theoretically possible that we might need a different number
| of different tools to take one apart? Sure but does that
| really count as a separate tool? And so it goes.. aren't we
| having fun yet??
| impendia wrote:
| I'd say one of the fundamental lessons of field and Galois
| theory is that they're _not_ intrinsically special. They 're
| just easier and more appealing to write down. (For the most
| part, anyway. They're a little bit special in some
| subspecialties for some technical reasons that are hard to
| explain.)
|
| One reason to focus on them: when you tell students that x^5
| - x - 1 = 0 can't be solved by radicals, that "even if God
| told you the answer, you would have no way to write it down",
| this is easy to understand and a powerful motivator for the
| theory. It's a nice application which is not fundamental, but
| which definitely shows that the theory has legs.
|
| If you want to know which polynomials are solvable by
| Newton's method? All of them. It illustrates that Newton's
| method is extremely useful, but the answer itself is not
| exactly interesting.
| returningfory2 wrote:
| > "even if God told you the answer, you would have no way
| to write it down"
|
| But isn't this also true for generic quadratics/cubics too?
| Like the solution to x^3-2=0 is cubed_root_of(2), so it
| seems we can "write it down". But what is the definition of
| cubed_root_of(2)? Well, it's the positive solution to
| x^3-2=0...
| impendia wrote:
| When I say that a fundamental lesson of field theory is
| that radicals are not really special, this is what I
| mean. You are thinking in a more sophisticated way than
| most newcomers to the subject.
| returningfory2 wrote:
| Oh sorry, right.
|
| I feel there is an interesting follow-up problem here.
| The polynomials x^n+a=0 are used to define the "radicals"
| which is a family of functions F_n such that F_n(a) =
| real nth root of a = real solution of x^n + a = 0. Using
| these radicals you can solve all quadratics, cubics and
| quintics.
|
| Now take another collection of unsolvable polynomials;
| your example was x^5 - x - 1 = 0 and maybe parameterize
| that in some way such that these polynomials are
| unsolvable. This gives us another family of functions
| G_n. What if we allow the G_n's to be used in our
| solutions? Can we solve all quintics this way (for
| example)?
| donkeybeer wrote:
| Bring radicals
| returningfory2 wrote:
| Awesome reference, thank you!
| impendia wrote:
| I'm not sure I understand your question exactly, but I am
| fairly certain that not all quintic fields can be solved
| by the combination of (1) radicals, i.e. taking roots of
| x^n - 1, and (2) taking roots of x^5 - x - 1. I don't
| have a proof in mind at the moment, but I speculate it's
| not too terribly difficult to prove.
|
| If I'm correct, then the proof would almost certainly use
| Galois theory!
| returningfory2 wrote:
| Thanks! I should have made it more explicit that we would
| need some family of quantic equations, not just x^5 - x -
| 1. And looks like from another reply the answer is yes? h
| ttps://en.wikipedia.org/wiki/Bring_radical#Solution_of_th
| e_...
| itishappy wrote:
| > Can we solve all quintics this way (for example)?
|
| Nope. Wanna hazard a guess what theory was instrumental
| in proving that there's no closed form for general
| quintics?
|
| https://en.wikipedia.org/wiki/Galois_theory#A_non-
| solvable_q...
| donkeybeer wrote:
| He is not wholly wrong, while not solvable in the general
| case by normal radicals, there is a family of functions,
| a special "radical" as he said, the Bring radical that
| solves the quintic generally. Of course as said its not a
| 5th root, but the solution to a certain family of
| quintics.
| itishappy wrote:
| Fascinating, thanks!
| enugu wrote:
| The prime importance of solving by radicals is actually, that
| it led to the theory of groups! Groups are used in all sorts
| of places. (One nice pictorial example is fundamental group
| of a topological space). Just like Complex numbers arose in
| trying to solve the cubic. Also, the statement of Fermat's
| Last Theorem doesn't have any applications but its solution
| led to lot of interesting theory like how ideals get
| factorized in rings, elliptic curves, Galois
| representations...
|
| BTW, the same theory can be extended to differential
| equations and Differential Galois Theory tells you if you can
| get a solution by composing basic functions along with
| exponentials.
|
| Historically, radicals can be motivated by looking at people
| trying to solve linear, then quadratics, medieval duels about
| cubics and quartics, the futile search for solving quintics
| etc. Incidentally, quintics and any degree can have a closed
| form solution using modular functions.
|
| More discussion on the MathOverflow page
| https://mathoverflow.net/questions/413468/why-do-we-make-
| suc...
| fredilo wrote:
| Superb question! Interestingly, my math professor asked the
| exact same question after teaching Galois theory and stated
| that he does not have a good answer himself. Let me try to
| give sort of an answer. :)
|
| We have fingers. These we can count. This is why we are
| interested in counting. This is gives us the natural numbers
| and why we are interested in them. What can we do with
| natural numbers? Well the basic axioms allow only one thing:
| Increment them.
|
| Now, it is a natural question to ask what happens when we
| increment repeatedly. This leads to addition of natural
| numbers. The next question is to ask is whether we can undo
| addition. This leads to subtraction. Next, we ask whether all
| natural numbers can be subtracted. The answer is no. Can we
| extend the natural numbers such that this is possible? Yes,
| and in come the integers.
|
| Now, that we have addition. We can ask whether we can repeat
| it. This leads to multiplication with a natural number. Next,
| we ask whether we can undo it and get division and rational
| numbers. We can also ask whether multiplication makes sense
| when both operands are non-natural.
|
| Now, that we have multiplication, we can ask whether we can
| repeat it. This gives us the raising to the power of a
| natural number. Can we undo this? This gives radicals. Can we
| take the root of any rational number? No, and in come
| rational field extensions including the complex numbers.
|
| A different train of thought asks what we can do with mixing
| multiplication and addition. An infinite number of these
| operations seems strange, so let's just ask what happens when
| we have finite number. It turns out, no matter how you
| combine multiplication and addition, you can always rearrange
| them to get a polynomial. Formulated differently: Every
| branch-free and loop-free finite program is a polynomial
| (when disregarding numeric stability). This view as a program
| is what motivates the study of polynomials.
|
| Now, that we have polynomials, we can ask whether we can undo
| them. This motivates looking at roots of polynomials.
|
| Now, we have radicals and roots of polynomials. Both
| motivated independently. It is natural to ask whether both
| trains of thought lead to the same mathematical object.
| Galois theory answers this and says no.
|
| This is a somewhat surprising result, because up to now, no
| matter in which order we asked the questions: Can we repeat?
| Can we undo? How to enable undo by extension? We always ended
| up with the same mathematical object. Here this is not the
| case. This is why the result of Galois theory is so
| surprising to some.
|
| Slightly off-topic but equally interesting is the question
| about what happens when we allow loops in our programs with
| multiplication and addition? i.e. we ask what happens when we
| mix an infinite number of addition and multiplication. Well,
| this is somewhat harder to formalize but a natural way to
| look at it is to say that we have some variable, in the
| programming sense, that we track in each loop iteration. The
| values that this variable takes forms a sequence. Now, the
| question is what will this variable end up being when we
| iterate very often. This leads to the concept of limit of a
| sequence.
|
| Sidenote: You can look at the usual mathematical limit
| notation as a program. The limit sign is the while-condition
| of the loop and the part that describes the sequence is the
| body of the loop.
|
| Now that we have limits and rational numbers, we can ask how
| to extend the rational numbers such that every rational
| sequence has a limit. This gives us the real numbers.
|
| Now we can ask the question of undoing the limit operation.
| Here the question is what undoing here actually means. One
| way to look at it is whether you can find for every limit,
| i.e., every real number, a multiply-add-loop-program that
| describes the sequence whose limit was taken. The answer
| turns out to be no. There is a countable infinite number of
| programs but uncountably infinite many real numbers. There
| are way more real numbers than programs. In my opinion this
| is a way stranger result than that of Galois theory. It turns
| out, that nearly no real number can be described by a
| program, or even more generally any textual description. For
| this reason, in my opinion, real numbers are the strangest
| construct in all of mathematics.
|
| I hope you found my rambling interesting. I just love to talk
| about this sort of stuff. :)
| glompers wrote:
| Yep!
| dataflow wrote:
| Thanks so much, I feel like you're the only one who grasps
| the crux of my question!
|
| > This is a somewhat surprising result, because up to now,
| no matter in which order we asked the questions: Can we
| repeat? Can we undo? How to enable undo by extension? We
| always ended up with the same mathematical object.
|
| I think this is the bit I'm confused on - we have an
| operation that is a mixture of two operations, where
| previously we only looked at pure compositions of
| operations (let's call this "impure"). Why is it surprising
| that the inversion of an "impure" operation produces an
| "impure" value?
|
| It's like saying, if I add x to itself a bunch, I always
| get a multiple of x. If I do the same thing with y, then I
| get a multiple of y. But if I add x and y to each other, I
| might get a prime number! Is that surprising? Mixing is a
| fundamentally new kind of operation; why would you expect
| its inversion to be familiar?
| fredilo wrote:
| What is surprising and what not is always very
| subjective.
|
| Now that I think more about it, one could argue that
| everything you can do with inverting radicals can also be
| done by inverting polynomials. So You could look at
| radicals as the step after multiplication and at
| inverting polynomials as the step after radicals. With
| this may depiction that these are two competing
| extensions falls apart a bit.
|
| My chain of argumentation was that one could expect that
| there is a single natural ever growing set of "numbers"
| starting with the natural numbers, then integers, then
| rational numbers, then real numbers, culminating in the
| real complex numbers and ever set is a superset of the
| previous one. This is the "natural" order in which they
| are taught in school and somewhat mirrors how they
| historically were discovered. In retrospect, this is
| obviously not true. Just look at the existence of
| rational complex numbers. However, when all you have are
| natural, integer and rational numbers, it seems like it
| could be true.
|
| Let me try a different way of explaining why it is
| surprising to some.
|
| In school, I learned that I can solve quadratic equations
| by combining the inverse operations of the basic
| operations that make up the quadratic polynom. This seems
| natural as it worked for solving the linear equations I
| had seen so far. Inverse of combination is combination of
| inverse. At some point the teacher showed the formula for
| degree three. Cubic radicals appeared. We were
| overwhelmed by it's size but the basic operations used
| matched what we expected. The teacher said that degree 4
| is even drastically larger with degree 4 radicals and we
| definitely do not want to see that, which is true.
| Nothing was said about degree 5 but it felt like it was
| implied that the pattern continues and the main problem
| with degree 5 is that our brains are just not able to
| handle the amount of operations that make up the formula.
|
| Fast forward to university. Now the professor proves in
| the Galois theory course that, no, it's not that you are
| too stupid to handle degree 5. It's just that degree 5
| cannot be handled this way at all. I am still unsure
| about whether my teacher in school knew that degree 5 is
| impossible or just assumed that he too is just too
| stupid.
|
| I guess this mathematicians must have felt something
| similar back then. You learn about linear equations. All
| is easy and works. You learn about quadratics. After
| mixing in quadratic radicals, all is well again. You try
| to grasp cubics, and yes, with a lot of work this too can
| be learned. You think about quartics and after lots and
| lots of time come to the conclusion that yes it is
| possible but impossible to master the formula. It feels
| like the pattern should continue and the reason you don't
| have a quintic formula with degree 5 radicals is not
| because it does not exist but because of it's sheer size
| and just stating it would fill a whole book. Turns out,
| there is no such book.
|
| Suppose you are a renowned mathematician back then who
| has failed for years to find a quintic formula. Now this
| teenager named Evariste comes along and fails too but
| says that it's not because he's too stupid but because
| it's impossible. At first, this does sound like an excuse
| of a lazy student, doesn't it?
|
| Let's say you are not surprised that roots of degree 5
| polynoms cannot be computed using just addition,
| subtraction, multiplication, division, and radicals. Does
| it surprise you that degree 4 polynoms can? Why does this
| work for degree 2, 3 and 4 yet fails for 5 and higher? I
| can see that one can argue that there is no reason to
| assume that it always works. However, at least learning
| the fact that it starts failing at degree 5 should be
| non-intuitive.
| lanstin wrote:
| They aren't really special except that adjoining the
| solutions to a radical to a field make the associated group
| of automorphisms simplify in a nice way. Also at the time
| tables of radical roots where common technology and Newton's
| method was not. But fundamentally, we can define and use new
| functions to solve things pretty easily; if you get down to
| it cos() and sin() are an example of this happening. All
| those applied maths weird functions (Bessel, Gamma, etc) are
| also this. As well, the reason not to just use numeric
| solutions for everything is that there are structural or
| dynamic properties of the solutions of equations that you
| won't be able to understand from a purely numeric
| perspective.
|
| I think taking a specific un-solvable numeric equation and
| deriving useful qualitative characteristics is a useful thing
| to try. You have cool simple results like Lyapunov stability
| criterion or signs of the eigenvalues around a singularity,
| and can numerically determine that a system of equations will
| have such and such long term behavior (or the tests can be
| inconclusive because the numerical values are just on the
| threshold between different behaviors.
|
| That's one of the really fun things about taking Galois
| theory class - you get general results for "all quintics" or
| "all quadratics" but also you can take specific polynomials
| and (sometimes) get concrete results (solvable by radicals,
| but also complex vs. real roots, etc.).
| sesm wrote:
| In a real problem you can have equations with parameters,
| where coefficients depend on some other parameters. And you
| should be able to answer questions about the roots depending
| on parameter values, like 'in which range of parameters p you
| have real (non-complex) roots'? Having a formula for the
| roots based on coefficients lets you answer such questions
| easily. For example, for quadratic equation ax^2 +bx + c = 0
| we have D = b^2 - 4ac, and if D < 0 then there is no real
| (non-complex) roots.
| nicf wrote:
| Lots of good answers here already, but I can also add my own
| perspective. Fundamentally, you're right that there isn't
| really anything "special" about radicals. The reason I
| personally find the unsolvability of the quintic by radicals
| interesting is that you _can_ solve quadratics, cubics, and
| quatrics (that is, polynomial equations of degree 2, 3, and
| 4) by radicals.
|
| To say the same thing another way: the quadratic formula that
| you learned in high school has been known in some form for
| millennia, and in particular you can reduce the question of
| solving quadratics to the question of finding square roots.
| So (provided you find solving polynomial equations to be an
| interesting question) it's fairly natural to ask whether
| there's an analogous formula for cubics. And it turns out
| there is! You need both cube roots and square roots, and the
| formula is longer and uglier, but that's probably not
| surprising.
|
| Whether you think n'th roots are "intrinsically" more
| interesting than general polynomial equations or not, this is
| still a pretty striking pattern, and one might naturally be
| curious about whether it continues for higher degrees. And I
| don't know anyone whose first guess would have been "yes, but
| only one more time, and then for degree 5 and higher it's
| suddenly impossible"!
| Koshkin wrote:
| I think it's quite natural to ask oneself how far one can go
| applying the same method that is used for solving the
| equation x2 = 4.
| Someone wrote:
| > To my programmer brain, whether I'm solving t^3 = 2 or
| t^100 + t + 1 = 7 numerically, I have to use an iterative
| method like Newton's either way.
|
| Doesn't your programmer brain want things to run as fast as
| possible?
|
| If you have another weapon in your arsenal for solving
| polynomial equations, you have an extra option for improving
| performance. As a trivial example, you don't call your Newton
| solver to solve a linear equation, as the function call
| overhead would mean giving up lots of performance.
|
| Also, if you solve an equation not because you want to know
| the roots of the equation but because you want to know
| whether they're different, the numerical approach may be much
| harder than the analytical one.
|
| > I would've thought I'd be more interested in whether
| they're solvable in some other respect (via iteration, via
| Newton, with quadratic convergence)
|
| That's fine. For that, you read up on the theory behind
| various iterative methods.
| cionx wrote:
| I think these are two different questions: - Why care about
| radicals? - Why try to solve polynomial equations in terms of
| radicals?
|
| For the first question:
|
| Taking Nth powers is a fairly basic operation, which occurs
| all the time in mathematics. Taking Nth roots is simply the
| inverse operation, so it is fairly natural to be interested
| in it/having to deal with it.
|
| For the second question:
|
| Let's pretend for a moment that we didn't know how the
| quadratic formula looked like. Could we nevertheless say
| anything about it?
|
| The quadratic formula is supposed to give us the solutions to
| the equation a x^2 + b x + c = 0. A special case of this
| general quadratic equation is x^2 - p = 0. There are two ways
| of solving this specialized equation: either by taking a
| square root, giving us the two solutions +-[?]p, or by using
| the general quadratic formula (with a = 1, b = 0, c = -p).
| Both of these approaches need to give us the same results,
| since they are both correct.
|
| This tells us that if we simplify the quadratic formula with
| a = 1, b = 0, c = -p, then a square root needs to appear. How
| can this happen? Well, the most basic guess is that the
| quadratic formula contained at least one square root to begin
| with.
|
| Looking at the actual quadratic formula tells us that this
| guess is correct: the formula uses the four basic arithmetic
| operations (addition, subtraction, multiplication, division)
| and a square root.
|
| We can repeat the same thought experiment for cubic
| equations, and we find that the cubic formula should probably
| contain third roots. Looking up the formula confirms this
| suspicion. However, it should be noted that the cubic
| equation does not only contain third roots, but also square
| roots.
|
| The situation for the quartic equation is similar: we suspect
| that the quartic formula contains fourth roots. And thanks to
| our experience with the cubic formula, we may also suspect
| that the quartic formula contains third roots and square
| roots. Looking up the formula, we see that it contains both
| third roots and square roots, but not (directly) any fourth
| roots. (Our original idea breaks down a bit because fourth
| roots can be expressed as iterated square roots. This makes
| it possible that the general quartic formula does not contain
| fourth roots, even though its simplified version will contain
| them.)
|
| So what about a general polynomial equations of degree N >=
| 5? Our original observation tells us that a solution formula
| needs to contain some sort of operation(s) that, when the
| formula is applied to certain special cases, gives us Nth
| roots. Just as before, the most basic guess is that the
| formula will contain Kth roots, and the previous examples
| suggest that one should expect K = 2, ..., N to occur.
|
| Summary: To find a formula for polynomials equations of
| degree N >= 2, we are _forced_ to use additional operations
| apart from the four basic arithmetic operations. In certain
| special cases, these additional operations _need_ to simplify
| to roots. This suggests using roots in the formula, and the
| cases N = 2, 3, 4 support this idea.
|
| Heuristically speaking, we are not trying to use roots
| because we want to, but because they seem to be the bare
| minimum required to even hope of finding a formula.
| tel wrote:
| You could think of it as a proof saying which polynomials are
| solvable by which algorithms. Solvable by radicals is one
| class of simpler algorithm and it so happens we have a cute
| proof as to when it will work or fail.
| atribecalledqst wrote:
| On the subject of more context in math, I've always wondered if
| having a grasp of the history of math would be helpful in
| getting better at solving mathematical problems. i.e. would
| learning more about how math developed over time, and how
| people solved important problems in the past, help me in trying
| to solve some other problem today?
|
| Years ago I bought the 3-volume set "Mathematical Thought from
| Ancient to Modern Times", but never had the time to get past
| the first few chapters. I'd be interested in any
| recommendations for math history tomes like that.
| mbivert wrote:
| > I'd be interested in any recommendations for math history
| tomes like that.
|
| Not a book, but FWIW, I've enjoyed a few videos from Norman
| Wildberger's "Math History" playlist[0]. Interestingly, he
| has a unconventional view of infinite processes in
| mathematics, a point of view that used to be common about a
| century ago or so.
|
| I'm sure knowing some amount of history is useful, but there
| must be a limit to how much of it is practically useful
| though.
|
| https://www.youtube.com/watch?v=dW8Cy6WrO94&list=PL55C7C8378.
| ..
| mindcrime wrote:
| I've been going through that very video lecture series the
| last couple of weeks. Good stuff. And in the lectures he
| mentions a number of books. I looked a few up on Amazon,
| and then looked at the associated Amazon recommendations,
| and so far have this small list of books related to Maths
| history that look worth reading:
| Mathematics and Its History (Undergraduate Texts in
| Mathematics) 3rd ed. 2010 Edition by John Stillwell
| The History of the Calculus and Its Conceptual Development
| (Dover Books on Mathematics) by Carl B. Boyer A
| History of Mathematics by Carl B. Boyer A
| Concise History of Mathematics: Fourth Revised Edition
| (Dover Books on Mathematics)A Concise History of
| Mathematics: Fourth Revised Edition (Dover Books on
| Mathematics) by Dirk J. Struik Introduction to
| the Foundations of Mathematics: Second Edition (Dover Books
| on Mathematics) Second Edition by Raymond L. Wilder
| Mathematical Thought from Ancient to Modern Times by Morris
| Kline (3 volume set) The Calculus Wars: Newton,
| Leibniz, and the Greatest Mathematical Clash of All Time by
| Jason Socrates Bardi
|
| There is also a "thing" in mathematics that is sometimes
| called the "genetic approach" where "genetic" is roughly
| equivalent to "historical" or maybe "developmental". IOW, a
| "genetic approach" book teaches a subject by tracing the
| development of the subject over its history. One popular
| book in this mold is: The Calculus: A
| Genetic Approach by Otto Toeplitz
| jaymzcampbell wrote:
| Stillwell's book is incredible, it hits the right balance
| between a textbook and a (advanced) lay person
| introduction to a huge range of topics.
| nyssos wrote:
| You're conflating a few things here.
|
| Constructivists are only interested in constructive proofs:
| if you want to claim "forall x in X, P(x) is true" then you
| need to exhibit a particular element of x for which P
| holds. As a philosophical stance this isn't super rare but
| I don't know if I would say it's ever been common. As a
| field of study it's quite valuable.
|
| Finitists go further and refuse to admit any infinite
| objects at all. This has always been pretty rare, and it's
| effectively dead now after the failure of Hilbert's
| program. It turns out you lose a ton of math this way -
| even statements that superficially appear to deal only with
| finite objects - including things as elementary as parts of
| arithmetic. Nonetheless there are still a few serious
| finitists.
|
| Ultrafinitists refuse to admit any _sufficiently large
| finite objects_. So for instance they deny that
| exponentiation is always well-defined. This is completely
| unworkable. It 's ultrafringe and always has been.
|
| Wildberger is an ultrafinitist.
| mbivert wrote:
| > You're conflating a few things here.
|
| It's likely: I purposefully stayed loose about the
| "infinite processes" to avoid going awry. I do however
| remembered him justifying his views as such though: he's
| not going into details, but he's making that point
| here[0] (c. 0:40). I assumed -- perhaps wrongfully --
| that he got those historical "facts" correct.
|
| https://youtu.be/I0JozyxM1M0?si=IFdWcEWNeNKDid7t&t=39
| kthielen wrote:
| > if you want to claim "forall x in X, P(x) is true" then
| you need to exhibit a particular element of x for which P
| holds
|
| I don't mean to be pedantic (although it's in keeping
| with constructivism) but in the case you describe, you
| don't have to provide a particular x but rather you have
| to provide a function mapping all x in X to P(x). It may
| very well be that X is uninhabited but this is still a
| valid constructive proof (anything follows from nothing,
| after all).
|
| If instead of "for all" you'd said "there exists", then
| yes constructivism requires that you deliver the goods
| you've promised.
| Tazerenix wrote:
| Wildberger is a crank
| mbivert wrote:
| A crank who provides hundreds of hours of fairly decent
| mathematical education content free of charge; it's not
| because he harbours unusual/fringe opinions that he's
| altogether worthless...
| kevindamm wrote:
| I found it helpful in some of my University math classes when
| I actually took the time to read the biographies that some of
| the textbooks included, the classes when I skimmed past them
| I did not remember details of the proof formulae for. But I
| don't know if this says more about the aid of history to the
| process of remembering or about my a priori interest in the
| topic for those particular classes.
|
| For a really good example of integrating the history along
| with the mathematics, and much more accessible than those
| math texts, I would recommend "Journey Through Genius" by
| Dunham[0]. It may be a little dated (published in 1990) and
| its focus is limited to algebra, geometry, number theory, and
| the history is perhaps too Western-biased, but it's good and
| it's short. Its material would make a solid foundation to
| build on top of because, in addition to the historical
| context, it shows a lot of the thought process into
| approaching certain landmark problems.
|
| [0]: https://www.goodreads.com/en/book/show/116185
| InitialLastName wrote:
| Boyer/Merzbach "A History of Mathematics" [0] is a tome in
| that vein. It spends a lot of time discussing the (mind-
| boggling, to a modern-mathematics-educated reader) methods
| that ancient peoples used to do real math (e.g. for
| engineering) as a way to motivate the development of the
| features of modern symbolic mathematics.
|
| [0] https://www.wiley.com/en-
| us/A+History+of+Mathematics%2C+3rd+...
| mamonster wrote:
| When I was an undergrad doing the mandatory measure theory
| course, I stumbled on a super old book(pre-1950 if I remember
| correctly, library card showed 3 people taking it out in the
| last 10 years) ,the name of which I forgot, that basically
| "re-built" the process that Lebesgue/Caratheodory/Riemann/etc
| followed, the problems they encountered (i.e Vitali set), why
| Lebesgue measure was the way it was and so on.
|
| I really wish I could remember the name of the book, but it
| made so much more sense than how even something like Stein
| Shakarchi or Billingsley, which introduced measures by either
| simply dumping the Vitali set on you as the main motivation
| or just not really explaining why stuff like outer
| measure/inner measure made sense.
| BeetleB wrote:
| > I found teaching Calculus I was able to draw in students so
| much more if I worked in what problems Newton was trying to
| solve and why. It gave them a story to follow, a reason to
| learn this stuff.
|
| Be careful.
|
| In undergrad, I worked as a math tutor at their tutoring
| institute. We had two calculus classes - one for
| engineers/science and the other for business/economics.
|
| If you're into Newtonian stuff or it's relevant to your degree,
| then your approach is all great. What I consistently saw is
| people in degrees like biology (or even industrial engineering)
| had a harder time learning _because_ of those physics
| applications in the book. They came in with one problem: Having
| trouble with the calculus. And then they discovered they had
| _two_ problems: To understand calculus they suddenly had to
| understand physics as well.
|
| You'd get students who were totally adept at differentiating
| and integrating, but would struggle with the problems that
| involved physics. Yes, it is important to be able to translate
| real world problems to math ones, but it's a bigger problem
| when you don't care about that particular field.
|
| Same problem with the business students. Since all the tutors
| were engineering folks, they had trouble tutoring the business
| students because their textbook was full of examples that
| required basic finance knowledge (and it really doesn't help
| that financial quantities are _discrete_ and not _continuous_ ,
| adding to the tutors' confusion).
|
| If you can find an application the student is interested in,
| then by all means, use that approach. For a general purpose
| textbook, though, it hinders learning for many students who
| don't care for the particular choice of application the book
| decided to use.
| dgacmu wrote:
| I got to see that yesterday with my daughter: her algebra
| book used PV=nRT as an example of joint proportion and asked
| some questions about it. She, having never seen the ideal gas
| law, was quite thrown by it.
|
| That said, after we went through it and had a brief physics
| lesson, it worked quite well and I'm glad they used the
| example instead of just making something up -- but it
| required having a tutor (me) on hand to help make the context
| make sense.
| HiroshiSan wrote:
| Ah yes I know that exact problem. AoPS intro to algebra.
| That question was great for getting an intuitive sense of
| proportion, though I do remember one of the sub problems
| gave me some trouble.
| dgacmu wrote:
| That's the one! It was a very nice example. I suspect for
| some students they could ignore the physics, but daughter
| needed to walk through the physical interpretation of the
| components before getting into the math.
|
| From my perspective as a tutor, it was a good use of time
| (gotta learn it some day anyway, and it provides useful
| physical intuition throughout life), but I could see it
| causing frustration if someone just wanted to learn
| algebra or didn't have a resource to turn to.
|
| (Love those books. I went and asked all of my colleagues
| who had won teaching awards, what books they recommended,
| and all of them said aops)
| arjvik wrote:
| I owe a great deal of my mathematical maturity to going
| through nearly every AoPS book published in middle and
| high school :)
| acchow wrote:
| > required having a tutor (me) on hand to help make the
| context make sense.
|
| I wonder how good ChatGPT would be as that tutor. You can
| ask it to "explain like I'm 12"
| CamperBob2 wrote:
| It's almost unbelievably good as a tutor. Still, you need
| to check _everything_ it tells you. Treat verification as
| part of the lesson.
| dgacmu wrote:
| So, I'm a professor, and I have, um, really strong
| opinions about this. :-) Perhaps too strong and long for
| the current forum. But I'll see if I can be brief.
|
| ChatGPT is really, really good at providing solid answers
| of varying levels of detail and complexity to hyper-
| common questions such as those used in problem sets. This
| is one part of the skill set of a tutor, and it's a
| valuable one.
|
| When I interview TAs for my classes, however, I actually
| put a lot more emphasis on a different skill: The ability
| to get into a student's head and understand where their
| conceptual difficulty or misunderstanding is. This is a
| very different skill, and it's one that ChatGPT isn't as
| good at, because we've gone from "maximum likelihood
| answers from questions that are in the middle of the
| distribution" into a wide range of possible sources of
| confusion, which the student may lack the words to
| explain in a precise way.
|
| In the case of my kid, the PV=nRT question manifested as
| "I don't get it!" (with more exclamation points).
|
| Asking ChatGPT (well, copilot, since I have institutional
| access to that, but it uses ChatGPT) to help understand
| the problem: It digressed and introduced Boyle's Law,
| threw in a new symbol "I" (ok, the 12 year old) had never
| seen for "proportional to", and ... in some sense just
| added to the cognitive overload.
|
| The human approach was to ask a question: Have you ever
| seen this equation before? (No) Oh! Well, let's talk a
| little about gases..
|
| Now, responding to ChatGPT and asking "No, that didn't
| help. Please ELI5 instead?" actually produced a much
| better answer: An analogy using a balloon. Which,
| amusingly, is exactly how I explained the behavior of
| gases to her.
|
| But even here, there's a bit of a difference: In
| explaining it to her, I did so socratically:
|
| "Ok, so imagine a balloon. If you heat the air inside the
| balloon, what happens?"
|
| "Um, it gets bigger, right?"
|
| "Yup, ..." (and now, knowing that she got that part, we
| could go on...)
|
| That's something you can _absolutely_ imagine trying to
| program around an LLM, but it 's not a native way of
| interacting with it.
|
| So ... I'd instead be a little more cautious here and say
| that ChatGPT potentially provides a really useful piece
| of what a human tutor offers, but it loses on the
| interactive exchange that helps much more rapidly zoom in
| on the source of confusion and correct it. Assuming that
| it's right.
|
| I think that for a particularly sophisticated consumer,
| it can be more valuable, but it requires knowing what you
| don't know, in some sense: The ability to already isolate
| what you're confused by. Once you know the question to
| ask, ChatGPT can often provide it -- again, assuming that
| some quirk of its training or your phrasing doesn't cause
| it to generate an answer that's wrong in some way.
| CamperBob2 wrote:
| I wonder if you are getting very different results with
| Copilot relative to what I've seen lately from ChatGPT4o.
| It did a pretty good job with the successive "Explain
| Galois theory like I am 5/16/133" prompts that I tried
| earlier. Seems likely that the chat system prompt is
| different from Copilot's, but who knows how.
|
| I will say that I have asked it to solve gas-law problems
| before, back when the base ChatGPT4 model was in use, and
| it didn't do so well there. It came up with the correct
| logic but I found it hard to get it to use consistent
| units, so the numbers kept coming out wrong.
|
| In general I find it super useful at improving my
| _conceptual_ understanding of things -- it 's at least as
| good as a good human tutor, at levels up to and including
| fairly-advanced signal processing topics -- but the
| details are less reliable. Feeding the same question to
| Gemini is a good way to double-check the numbers in my
| experience. If they agree, the numbers are probably
| right, while if they don't, then it's likely they're both
| wrong.
| silasdavis wrote:
| Can you ask it to explain galois theory like he's 21?
| CamperBob2 wrote:
| ----------------
|
| "Let's imagine Evariste Galois survived that fateful duel
| and continued his work. What would he have done next? If
| Galois had lived past 21, he would have likely expanded
| and deepened his revolutionary ideas. By that time, he
| had already laid down the foundation of what we now call
| Galois theory, but there were still many directions he
| could have explored further.
|
| _Generalization of Galois Theory_ : Galois had just
| scratched the surface of the powerful connection between
| field theory and group theory. His next logical step
| might have been to generalize his ideas to more complex
| situations, such as understanding the symmetries of
| solutions to equations in multiple variables or extending
| his work to non-polynomial equations.
|
| _Exploration of Solvable Groups_ : Galois introduced the
| concept of solvable groups to determine whether a
| polynomial equation could be solved by radicals (using
| addition, multiplication, and roots). If he had lived
| longer, he would likely have delved deeper into
| classifying different types of groups and exploring their
| properties. This could have led to a more detailed theory
| of group structures, potentially anticipating parts of
| what we now know as modern group theory.
|
| _Linking Galois Theory with Other Areas_ : Galois might
| have explored connections between his theory and other
| areas of mathematics, such as geometry, number theory, or
| even early versions of what we now call topology. For
| instance, he could have investigated how the symmetries
| of geometric shapes relate to the symmetries of algebraic
| equations.
|
| _Developing Further Applications_ : Beyond theory,
| Galois could have sought applications of his work in more
| practical areas. Perhaps he would have applied his ideas
| to solving problems in physics, mechanics, or other
| emerging fields, where symmetry plays a crucial role.
|
| _Inspiring the Next Generation_ : If Galois had lived,
| he might have become a prominent teacher and mentor,
| influencing a new generation of mathematicians. His
| radical ideas would have challenged and inspired others
| to push the boundaries of mathematics even further.
| Galois could have spearheaded a new movement in algebra,
| similar to how Einstein revolutionized physics.
|
| _Refining His Work_ : Galois' original manuscripts were
| hastily written under dire circumstances. Given more
| time, he would likely have polished and clarified his
| ideas, making them more accessible to other
| mathematicians of the time. This might have sped up the
| acceptance and expansion of his ideas in the mathematical
| community."
|
| ----------------
|
| I'd give it a C for effort, I guess. I don't know enough
| about the topic to ask it for more specific predictions.
| At the end of the day it can only work with what it was
| trained with, so any genuine insights it comes up with
| will be due more to accident than inspiration.
| magicalhippo wrote:
| > The ability to get into a student's head and understand
| where their conceptual difficulty or misunderstanding is.
|
| I experienced that first hand as someone who just enjoyed
| math and had several courses from uni, and tried to help
| my SO and a few friends which struggled hard with
| different pre- or entry-level college math courses. They
| all needed quite different approaches to be able to
| understand the material.
|
| For one I had to go all the way back and re-learn basic
| algebra as they had had a poor teacher which hadn't
| properly taught that. It would manifest in not
| understanding steps, not being able to solve equations
| properly and so on.
|
| One really didn't get the visual graph explanation of
| derivation of composite functions, and instead got it by
| deriving the formula and using it in several examples. An
| approach which didn't work with the others as they needed
| the graph as a reference or motivation.
|
| Was a very interesting experience, and as you say a very
| different challenge from just knowing the source material
| well.
| jacobolus wrote:
| ChatGPT is an unbelievably _bad_ tutor if what you want a
| tutorial about is even a little bit obscure (e.g. the
| answer you want isn 't already included in Wikipedia). It
| just confidently states vaguely plausible sounding made
| up nonsense, and then when you ask it if it was mistaken
| it shamelessly makes up different total nonsense, and if
| you ask it for sources it makes up non-existent sources,
| and then when you look whatever it was up for yourself
| you have to spend 2x as long as you originally would have
| chasing down wrong paths and trying to understand exactly
| which parts ChatGPT was wrong about (most of them).
|
| And that's assuming you are a very savvy and media
| literate inquirer with plenty of domain expertise.
|
| In cases where the answer you want was already easily
| findable, ChatGPT still is wrong about a lot of it, and
| you could have more easily gotten a (mostly) correct
| answer by looking at standard sources, or if you want to
| be more careful tracking down _their_ actually existing
| cited sources or doing a skim search through the academic
| literature.
|
| If you ask it something in a topic you are not already an
| expert about, or if you are e.g. an ordinary high school
| or college student, you are almost certainly coming away
| from the conversation with serious misconceptions.
| CamperBob2 wrote:
| _ChatGPT is an unbelievably bad tutor if what you want a
| tutorial about is even a little bit obscure_
|
| That has absolutely not been my experience at all. It's
| brought me up to speed in areas from ML to advanced DSP
| that I'd been struggling with for a long time.
|
| How long has it been since you used it, and what did you
| ask it?
| jacobolus wrote:
| I have tried asking it all sorts of questions about
| specific obscure word etymologies and translations,
| obscure people's biographies (ancient and modern),
| historical events, organizations, academic citations,
| mathematical definitions and theorems, physical
| experiments, old machines, native plants, chemical
| reactions, diseases, engineering methods, ..., and it
| almost invariably flubs every question I throw at it,
| sometimes subtly and sometimes quite dramatically, often
| making up abject nonsense out of whole cloth. As a result
| I don't bother too much; I've found it to waste more time
| than it saves. To be fair, the kinds of questions I would
| want a tool like this to answer are usually ones I would
| have to spend some time and effort hunting to answer
| properly, and I'm pretty fast and effective at finding
| information.
|
| I haven't tried asking too much about questions that I
| could trivially answer some other way. If what you want
| to know can be found in any intro undergrad textbook or
| standard dictionary (or Wikipedia), it's plausible that
| it would be better able to parrot back more or less the
| correct thing. But again, I haven't done much of this,
| preferring to just get hold of the relevant dictionary or
| textbook and read it directly.
|
| I'll give you an example. I just now asked chatgpt.com
| what Lexell's theorem is and it says this:
|
| > _Lexell 's theorem is a result in geometry related to
| spherical triangles. Named after the mathematician Michel
| Leonard Jean Leclerc, known as Lexell, it states: P In a
| spherical triangle, if the sum of the angles is greater
| than p radians (or 180 degrees), then the spherical
| excess (the amount by which the sum of the angles exceeds
| p) is equal to the area of the spherical triangle on a
| unit sphere. P In simpler terms, for a spherical
| triangle, the difference between the sum of its angles
| and p radians (180 degrees) gives the area of the
| triangle when the sphere is of unit radius. This theorem
| is fundamental in spherical geometry and helps relate
| angular measurements directly to areas on a sphere._
|
| This gets the basic topic right ("is a result in geometry
| related to spherical triangles", involves area or
| spherical excess) but everything else about the answer,
| starting with the mathematician's identity, is completely
| wrong.
|
| If I tell it that this is incorrect, it repeats a random
| assortment of other statements, none of which is actually
| the theorem I am asking about. E.g.
|
| > [...] _In a spherical triangle, if you have a spherical
| triangle with vertices A, B, and C, and the sides of the
| triangle are a, b, and c (measured in radians), then: P
| cos[?](a)cos[?](b) + sin[?](a)sin[?](b)cos[?](C) =
| cos[?](c)._ [...]
|
| or
|
| > [...] _In a spherical polyhedron, the sum of the angles
| at each vertex is equal to 2p radians minus the sum of
| the interior angles of the faces meeting at that vertex._
| [...]
|
| If you want to know what Lexell's theorem actually is,
| you can read the Wikipedia article I wrote last year:
| https://en.wikipedia.org/wiki/Lexell%27s_theorem
|
| > _every spherical triangle with the same surface area on
| a fixed base has its apex on a small circle, called
| Lexell 's circle or Lexell's locus, passing through each
| of the two points antipodal to the two base vertices._
|
| The problem ChatGPT has is that it's not able to just say
| something true but incomplete such as "I'm not sure what
| Lexell's theorem is or who Lexell was, but I know the
| theorem has something to do with spherical trigonometry;
| maybe it could be found in the more comprehensive books
| about the subject such as Todhunter & Leathem 1901 or
| Casey 1889".
|
| Instead it just authoritatively spouts one bit of
| nonsense after another. (Every topic I have ever tried
| asking it about in detail is more or less the same.) The
| incorrect statements range from subtly wrong (e.g. two
| different things with similar names got conflated and
| some of the properties of the more common one were
| incorrectly applied to the other) to complete nonsense
| (jumbles of technical jargon strung together that are
| more or less gibberish). It's clear if you read carefully
| about any technical topic that it doesn't actually
| understand what it is saying, and is just combining bits
| of vaguely related material. Answers to technical
| questions are almost never entirely technically accurate
| unless you ask a very standard question about a very
| basic topic.
|
| Anyone using it for any purpose should (a) be already
| pretty media literate with some domain expertise, and (b)
| be willing to carefully verify every part of every
| statement.
| CamperBob2 wrote:
| Can't argue with that. Your earlier point is the key:
| "e.g. the answer you want isn't already included in
| Wikipedia." Anything specialized enough not to be covered
| by Wikipedia or similar resources -- or where, in your
| specific example, the topic was only recently added -- is
| not a good subject for ChatGPT. Not yet, anyway.
|
| Now, pretend you're taking your first linear algebra
| course, and you don't quite understand the whole
| determinant thing. Go ask it for help with _that_ , and
| you will have a very different experience.
|
| In my own case, what opened my eyes was asking it for
| some insights into computing the Cramer-Rao bound in
| communications theory. I needed to come up to speed in
| that area awhile back, but I'm missing some prereqs, so
| textbook chapters on the topic aren't as helpful as an
| interactive conversation with an in-person tutor would
| be. I was blown away at how effective GPT4o was at
| answering follow-up questions and imparting actionable
| insights.
| kergonath wrote:
| A problem, though, is that it is not binary. There is a
| whole spectrum of nonsense, and if you are not a
| specialist it is not obvious to figure out the accuracy
| of the reply. Sometimes by chance you end up asking for
| something the model knows about for some reason, but very
| often not. That is the wrong aspect of it. Students might
| rely on it in their 1st year because it worked a couple
| of times and then learn a lot of nonsense among the
| truthy facts LLMs tend to produce.
|
| The main problem is not that they are wrong. It would be
| simpler if they were. But then, recommending students to
| use them as tutors is really not a good idea, unless what
| you want is overconfidently wrong students (I mean more
| than some of them already are). It's not random
| doomsayers saying this; it's university professors and
| researchers with advanced knowledge. Exactly the people
| that should be trusted for this kind of things, more than
| AI techbros.
| jacobolus wrote:
| > _Anything specialized enough not to be covered by
| Wikipedia or similar resources [...] is not a good
| subject for ChatGPT._
|
| Things don't have to be incredibly obscure to make
| ChatGPT completely flub them (while authoritatively
| pretending it knows all the answers), they just have to
| be slightly beyond the most basic details of a common
| subject discussed at about the undergraduate level.
| Lexell's theorem, to take my previous example, is
| discussed in a wide variety of sources over the past 2.5
| centuries, including books and papers by several of the
| most famous mathematicians in history, canonical
| undergraduate-level spherical trigonometry textbooks from
| the mid 20th century, and several easy-to-find papers
| from the past couple decades, including historical and
| mathematical surveys of the topic. It just doesn't happen
| to be included in the training data of reddit comments
| and github commit messages or whatever, because it
| doesn't get included in intro college courses so nobody
| is asking for homework help about it.
|
| If you stick to asking single questions like "what is
| Pythagoras's theorem" or "what is the most common element
| in the Earth's atmosphere" or "who was the 4th president
| of the USA" or "what is the word for 'dog' in French",
| you are fine. But as soon as you start asking questions
| that require knowledge beyond copy/pasting sections of
| introductory textbooks, ChatGPT starts making (often
| significant) errors.
|
| As a different kind of example, I have asked ChatGPT to
| translate straightforward sentences and gotten back a
| translation with exactly the opposite meaning intended by
| the original (as verified by asking a native speaker).
|
| The limits of its knowledge and response style make
| ChatGPT mostly worthless to me. If something I want to
| know can be copy/pasted from obvious introductory
| sources, I can already find it trivially and quickly. And
| I can't really trust it even for basic routine stuff,
| because it doesn't link to reliable sources which makes
| its claims unnecessarily difficult to verify. Even
| published work by professionals often contains factual
| errors, but when you read them you can judge their
| name/reputation, look at any cited sources, compare
| claims from one source to another, and so on. But if
| ChatGPT tells you something, you have no idea if it read
| it on a conspiracist blog, found it in the canonical
| survey paper about the topic, or just made it up.
|
| > _Go ask it for help [understanding determinants], and
| you will have a very different experience._
|
| It's going to give you the right basic explanation (more
| or less copy/pasted from some well written textbook or
| website), but if you start asking follow-up questions
| that get more technically involved you are likely to hit
| serious errors within not too many hops which reveal that
| it doesn't actually understand what a determinant is, but
| only knows how to selectively regurgitate/paraphrase from
| its training corpus (and routinely picks the wrong source
| to paraphrase or mashes up two unrelated topics).
|
| You can get the same accurate basic explanation by doing
| a quick search for "determinant" in a few introductory
| linear algebra textbooks, without really that much more
| trouble; the overhead of finding sources is small
| compared to the effort required to read and think about
| them.
| acchow wrote:
| Are you using the free version? GPT 4 Turbo (which is
| paid) gives this:
|
| > Lexell's theorem is a result in geometry related to
| triangles and circles. Named after the mathematician
| Anders Johan Lexell, the theorem describes a special
| relationship between a triangle and a circle inscribed in
| one of its angles. Here's the theorem:
|
| Given a triangle \\(ABC\\) and a circle that passes
| through \\(B\\) and \\(C\\) and is tangent to one of the
| sides of the angle at \\(A\\) (say \\(AB\\)), the theorem
| states that the circle's other tangent point with
| \\(AB\\) will lie on the circumcircle of triangle
| \\(ABC\\).
|
| In other words, if you have a circle that touches two
| sides of a triangle and passes through the other two
| vertices, the point where the circle touches the third
| side externally will always lie on the triangle's
| circumcircle. This theorem is useful in solving various
| geometric problems involving circles and triangles.
| wizzwizz4 wrote:
| > _It 's brought me up to speed in areas from ML to
| advanced DSP that I'd been struggling with for a long
| time._
|
| Are you sure it did? Or did it just convince you that you
| understood it?
| CamperBob2 wrote:
| If the code I wrote based on my newly-acquired insight
| works, which it does, that's good enough for me.
|
| Beyond that, there seems to be some kind of religious war
| in play on this topic, about which I have no opinion...
| at least, none that would be welcomed here.
| wizzwizz4 wrote:
| ML and DSP are both areas where buggy code _seems_ to
| work, but actually gives suboptimal performance / wrong
| results. See:
| https://karpathy.github.io/2019/04/25/recipe/#2-neural-
| net-t...
|
| > The "possible error surface" is large, logical (as
| opposed to syntactic), and very tricky to unit test. For
| example, perhaps you forgot to flip your labels when you
| left-right flipped the image during data augmentation.
| Your net can still (shockingly) work pretty well because
| your network can internally learn to detect flipped
| images and then it left-right flips its predictions. Or
| maybe your autoregressive model accidentally takes the
| thing it's trying to predict as an input due to an off-
| by-one bug. Or you tried to clip your gradients but
| instead [...]
|
| > Therefore, your misconfigured neural net will throw
| exceptions only if you're lucky; Most of the time it will
| train but silently work a bit worse.
| CamperBob2 wrote:
| Actually Karpathy is a good example to cite. I took a few
| months off last year and went through his "Zero to hero"
| videos among other things, following along to reimplement
| his examples in C++ as an introductory learning exercise.
| I spent a lot of time going back and forth with ChatGPT
| to understand various aspects of backpropagation through
| operations including matmuls and softmax. I ended up well
| ahead of where I would otherwise have been, starting out
| as a rank noob.
|
| Look: again, this is some kind of religious thing where a
| lot of people with vested interests (e.g., professors)
| are trying to plug the proverbial dyke. Just how much
| water there is on the other side remains to be seen. But
| finding ways to trip up a language model by challenging
| its math skills isn't the flex a lot of you folks think
| it is... and when you discourage students from taking
| advantage of every tool available to them, you aren't
| doing them the favor you think you are. AI got a hell of
| a lot smarter over the past few years, along with many
| people who have found ways to use it effectively. Did
| you?
|
| With regard to being fooled by buggy code or being
| satisfied with mistaken understanding, you don't know me
| from Adam, but if you did you'd give me a _little_ more
| credit than that.
| BeetleB wrote:
| Now imagine you were reading a mathematics textbook, and
| they use an example from something you really have no
| interest in (which for me would be finance). It includes
| terminology you have to learn - terminology that has no
| bearing on the math.
|
| As discouraging it is to learn math without context, it's
| even more discouraging to learn it in a context you hate.
| Koshkin wrote:
| I have no interest in finance but I liked this book:
|
| https://nononsense.gumroad.com/l/physicsfromfinance
| freestyle24147 wrote:
| If someone is having trouble understanding the absolute most
| basic part of classical mechanics (masses undergoing
| acceleration) something tells me they're not going to
| understand Calculus no matter what lens you view it through.
| It's not even about the equations, it's simply using it as a
| backstory for the motivation of the mathematics.
|
| > If you can find an application the student is interested
| in, then by all means, use that approach. For a general
| purpose textbook, though, it hinders learning for many
| students who don't care for the particular choice of
| application the book decided to use.
|
| This seems absurd. Just because some people will find a
| particular application less interesting, I don't think the
| answer is to throw out ALL applications and turn it into a
| generic boring slog through which no one will be able to see
| when and how it's useful.
| VyseofArcadia wrote:
| > It's not even about the equations, it's simply using it
| as a backstory for the motivation of the mathematics.
|
| Bingo. I rarely (if ever) used motivating examples from
| physics as test or homework problems. This was to ground
| calculus in reality somewhere. This was after years of
| wondering how to deal with the common student complaint of,
| "but why, where does this even come from?"
|
| So I started telling them where it comes from.
| calf wrote:
| The alternative is to ground it in philosophy and
| theoretical mathematics which would be even more
| abstruse...
| acchow wrote:
| Most people don't need or care to learn calculus. They just
| want to learn how to _use calculus_
| gowld wrote:
| Similar to how most people don't need or care to learn
| statistics. They just want to learn how to say
| statistical words to silence critics of their
| pseudoscience.
| BeetleB wrote:
| > something tells me they're not going to understand
| Calculus no matter what lens you view it through.
|
| From my experience as a tutor, you are quite wrong in
| coming to that conclusion. Most people conflate mass and
| weight all the time, and have a fuzzy understanding of
| acceleration. It's not because they're thick in the head
| and incapable, but because _they don 't care_. That doesn't
| mean they don't care about other applications where math
| can help.
|
| And standard calculus textbooks go beyond what you are
| describing. All the ones I encountered would have the
| integral of force with displacement to get work (which
| confuses people when they hear it's the same as "energy").
|
| > This seems absurd.
|
| The truth often does seem so.
| jbaber wrote:
| This is spot on. Everybody has lots of favorite subjects.
| They're not all the same.
| crazygringo wrote:
| > _Most people conflate mass and weight all the time, and
| have a fuzzy understanding of acceleration._
|
| Conflating mass and weight is generally irrelevant in
| calculus textbooks, since they're generally giving you
| the mass, and weight doesn't even come up.
|
| And coming in having a fuzzy understanding of
| acceleration is fine, _because calculus is where you
| learn what acceleration is._
|
| Learning that velocity and acceleration are the first and
| second derivatives _is_ the most intuitive way to
| introduce them to anyone.
|
| If you're taking calculus but you don't want to learn
| what acceleration is, then I don't know what you're even
| doing. Even if you're doing it for finance or medicine or
| something, velocity and acceleration are still the most
| useful and intuitive ways to introduce derivatives.
| AdamN wrote:
| You're missing the part that they may have taken zero
| classical mechanics classes. Even if it's something you
| learn in week 3 in that class, if you've never taken it you
| have no idea where those ideas fit in.
| gowld wrote:
| Why is someone studying science in college, if they
| haven't finished high school?
| ordu wrote:
| I'm now learning German in Duolingo, and I noticed a
| correlation between how hard it is for me to learn one more
| pack of a vocabulary and how this vocabulary is relevant
| for me.
|
| There is nothing that I cannot understand, but when some
| words do not relate to my needs I need to make a conscious
| effort to learn them. When the words are the ones I'd
| happily use, I'll learn them easily without any effort, I
| just need to spend enough time with Duolingo.
|
| I believe that people who do not care about physics are
| having the very same issues with calculus that was
| explained with references to physics. Probably they can
| overcome this difficulties, like I can overcome my
| difficulties with uninteresting words, but it means they
| need to spend more effort and more time to get the same
| result.
| mihaitodor wrote:
| That's me with statistics. I have 0 interest in gambling and
| sports, so most of the examples put me off instantly. I'll
| need to study this topic in the context of biology
| (bioinformatics), so I'll want to find a good stats learning
| guide which avoids those topics.
| mp05 wrote:
| Your mileage may vary, but my industrial engineering program
| had two semesters of proper physics and two semesters of
| engineering mechanics.
| chrisweekly wrote:
| Yeah!
|
| You're probably already aware of https://betterexplained.com --
| an amazing resource that exemplifies this same mindset.
| VyseofArcadia wrote:
| I was not, thank you for this.
| alejohausner wrote:
| When I took real analysis, I didn't get an intuitive sense of
| limits and all that delta/epsilon stuff. It was too abstract.
| Strangely enough, when I read David Foster Wallace's
| "Everything and More" on the history of infinity it all made
| more sense, because he described all the paradoxes and dead
| ends that mathematicians had run into before Cauchy and others
| brought rigor to the problem. Wallace was a postmodern
| novelist. I don't know why he described infinity so clearly; I
| didn't enjoy his other work.
| Koshkin wrote:
| Interesting... This e-d stuff is needed when you talk about
| functions, but when you start instead with the limits of
| numeric sequences, which are easy to grasp, the function
| limits come easy as well, because of the analogy between the
| ways these are defined.
| alejohausner wrote:
| It's not so much that I didn't understand the principles
| behind continuity. It's just that the way my teachers
| presented the material lacked historical context.
|
| After a BSc in pure math I discovered that I enjoyed
| applied math and CS much more, which told me that I need
| concrete examples to understand a theory: if you tell me
| about abstractions like groups and rings, which took years
| to establish, I lose interest. Tell me that groups express
| properties of matrix multiplications, or permutations, or
| modular arithmetic, and I'll get it right away.
|
| It's the way my mind works, but I'm sure I'm not alone, and
| mathematical pedagogy would benefit from historical
| context.
| silasdavis wrote:
| The maths appendices in Infinite Jest are about half the
| book. There's a section written by Hal's friend (?), proving
| something like the intermediate value theorem where he says
| something like: we'll use epsilon delta because it's mad fun
| to say.
| ForOldHack wrote:
| Chapter X is brilliant. Galois was very brilliant. He is always
| my #1 choice for ever of who, if anyone from history, I would
| like to have lunch with.
|
| He was introduced to me by my 5th grade calculus teacher Mr
| Steven Giavant, PhD. Galois bridged several disciplines and
| invented a new area of math.
|
| Most unfortunately he met his end extremely tragically.
|
| https://en.wikipedia.org/wiki/%C3%89variste_Galois
| tzs wrote:
| > He was introduced to me by my 5th grade calculus teacher
| [...]
|
| 5th grade calculus!?
| XMPPwocky wrote:
| Absolutely agree. When I was a kid, I took a MOOC (MITx 6.002x)
| because I was interested in EE. They said calculus was a
| prerequisite, which I didn't know yet, but I went ahead
| anyways; every time the course required calculus, I'd go read
| the relevant chapters in Strang's _Calculus_. Felt incredibly
| natural, and I ended up going through the rest of Strang just
| for fun- probably the best learning experience I 've ever had,
| and I doubt I'd have ever done it without having the MOOC there
| to motivate the problem.
| raegis wrote:
| > I've been shouting from the rooftops for years that math[0]
| courses need more context.
|
| A slightly different perspective: It used to be common for
| professors to teach a lot of history in graduate algebra
| courses. (Generation X and older.) I've talked to students from
| a few other schools who experienced this. None of us cared for
| it at the time. My teacher spent at least two class periods
| writing down the history on the board. We were all quite bored.
| But then again, we were in math grad school, so there was no
| need to try to motivate us.
| gowld wrote:
| Notes, Videos, and Problems:
| https://www.maths.ed.ac.uk/~tl/galois/#notes
|
| Direct link to PDF of notes: https://arxiv.org/pdf/2408.07499
| mkw5053 wrote:
| A few years ago, I led a study group through A Book of Abstract
| Algebra by Charles C Pinter. It culminated in Galois Theory and
| was one of the best books I've ever used in a math study group.
| mkw5053 wrote:
| Also, it assumes little to no advanced math knowledge and can
| be found for free online :)
| sifar wrote:
| I second this book, really accessible. It taught me Abstract
| Algebra when I was learning it by myself.
| bosquefrio wrote:
| I have fond memories of reading this book in college. I enjoyed
| it immensely.
|
| I also remember reading the section at the back of the book
| about Galois. There was also an entertaining section about the
| history of solving the roots of polynomial equations and in
| particular solving equations of arbitrary order.
| PreInternet01 wrote:
| [removed by author]
| ogogmad wrote:
| I don't think this is Galois theory. Galois theory is about
| "The Fundamental Theorem of Galois Theory", which states that
| there is a nice mapping from field-extensions to group-
| extensions, where the resulting groups are usually finite. When
| the resulting groups are finite (as they usually are), many
| problems involving field extensions can be solved using brute-
| force search.
|
| _Galois fields_ happen to be something else named in honour of
| Galois.
| koolala wrote:
| Galois Fields yeah. Being able to store Rational Numbers in a
| computer.
|
| "Originally, the theory had been developed for algebraic
| equations whose coefficients are rational numbers."
| matt-noonan wrote:
| That is definitely not what Galois Fields are about.
| koolala wrote:
| https://en.m.wikipedia.org/wiki/Galois_theory#Permutation
| _gr...
|
| That quote about equations with rational numbers was from
| here. Galois didn't have a computer of course. Rational
| Numbers and trisecting an angle sound related.
| wging wrote:
| The quote is followed immediately by this: "It extends
| naturally to equations with coefficients *in any field*,
| but this will not be considered in the simple examples
| below." Emphasis on 'in any field' is mine. Among the
| other fields that can be considered include the Galois
| fields, which are another name for finite fields. (There
| are also infinite fields other than the rationals, so 'in
| any field' does not _just_ mean Galois fields /finite
| fields.) https://en.wikipedia.org/wiki/Finite_field
|
| Galois fields have nothing to do with being able to
| represent rational numbers in a computer: elements of a
| finite field aren't even rational numbers.
| koolala wrote:
| I can see how the idea extends naturally and also how it
| doesn't extend naturally. Thanks for explaining, I wonder
| what Galois would do if alive today with computers.
| graemep wrote:
| Patent threats can be used to stop something being taught,
| rather than being used? That is awful.
| koolala wrote:
| My memory of the story was they got some research funding
| from the troll company and were naive about human greed.
| koolala wrote:
| I had no idea Plank's work connected to Galois. I wish a
| University was able to publish free open information but money
| decides research.
| ogogmad wrote:
| Interestingly, there's a close connection between the
| "Fundamental Theorem of Galois Theory" and the "Fundamental
| Theorem of Covering Spaces".
| HPsquared wrote:
| For non-math people, is this "simple Wikipedia" article about
| right? I've always seen Galois theory listed in mathematics
| courses and wondered what it is, speaking as a humble engineer.
| https://simple.m.wikipedia.org/wiki/Galois_theory
| gnulinux wrote:
| Yes and no, it's a bit too simplistic and doesn't explain the
| actual "why" of Galois Theory, just the how. The brilliant
| insight Galois figured out is that there is a fundamental
| connection between fields and groups, but that is just the
| "technique" with which he solved the problem. The "why even
| bother" is a bit more complex but simply put Galois wanted to
| establish a criterion to determine what polynomials are
| solvable or unsolvable in which fields. E.g. we know x^2=-1 is
| solvable in C with x=i but not in real numbers. Can we
| generalize that proof to such a degree that we can
| mechanistically run it for arbitrary polynomials in arbitrary
| fields?
| fredilo wrote:
| What it states is correct and it gives you a good overview over
| what you do in a Galois theory course. It does, however, not
| give you an idea of why this is interesting. When just reading
| that article one might get the idea that some mathematicians
| just had too much free time.
|
| I tried to motivate the questions leading to Galois Theory in
| https://news.ycombinator.com/item?id=41258726 in a way that is
| hopefully accessible to more down-to-earth programmers and
| engineers.
| fredilo wrote:
| I should probably add why I think the motivation is so
| important here. For pure engineers, numbers are a tool. They
| ask: What can I build with numbers? Pure mathematicians ask a
| different question. They are interested in the limits of
| numbers. They ask: What can I not build with numbers?
| Studying these two questions is deeply related but also a
| constant source of frustration for engineers taking math
| courses designed for mathematicians by mathematicians.
|
| Galois theory, is a theory of "no". It ultimately serves to
| answer several "Can I build this?" questions with no. This
| makes it very interesting to pure mathematicians. However,
| for pure engineers that are looking for numeric machine parts
| that can be assembled in other useful ways to actually build
| something... Galois theory can be quite disappointing.
| klyrs wrote:
| Welcome to HN! FYI, you can edit comments for two hours
| after posting them.
| raldi wrote:
| Clicked around for a few minutes and couldn't find a sentence
| beginning, "Galois Theory is..."
| chii wrote:
| https://www.maths.ed.ac.uk/~tl/galois/
|
| it's a link to the course, and there's an introduction right at
| the beginning of the course.
|
| Unfortunately, it's a video, rather than text, but there's at
| least transcript.
|
| Even wikipedia acknowledges that it is not a simple subject:
| https://simple.wikipedia.org/wiki/Galois_theory#Disclaimer ;so
| hoping you can get an understanding of it in a few sentences is
| probably asking too much tbh.
| skulk wrote:
| Someone could tell you that Galois theory is the study of field
| extensions and the structure of their automorphism groups but
| is that really going to help?
| gowld wrote:
| The fundamental theorem of Galois theory reduces certain
| problems in field theory (like finding roots of polynomials)
| to group theory (counting permutations and symmetries), which
| makes them simpler and easier to understand and solve (or
| prove non-solvable). Galois theory is the proof and
| applications of this theorem, and related topics. Key to this
| theory is being more methodical about extending the rational
| numbers into the real numbers, by introducing new numbers one
| at a time, instead of all at once.
|
| One of the immediate discoveries in beginning this study is
| the fact that in many common cases you cannot add just one
| numbers one at a time, but must add 2 or more numbers at
| once. These sets of numbers are called conjugates, and have
| the interesting property that even though you can prove how
| many must exist and that they are distinct from each other,
| they are otherwise identical except in the arbitray names you
| give them.
| taneliv wrote:
| I watched one of the videos, titled "Galois groups,
| intuitively".[1] It is about 18 minutes long and gave me a
| somewhat understandable overview what Galois groups are (not
| the theory yet!).
|
| I suppose if you have the time to spare, at least it should
| give you an idea if the topic is of interest and whether you
| need to remind yourself of some mathematical concepts to be
| able to study rest of the material, or if it is too elementary
| for you and a shorter treatise from elsewhere would be
| preferable.
|
| (I actually thought I had learned something about Galois theory
| in University algebra, but either I hadn't, or I've forgotten
| more than I wanted to admit. Which is to say, if you watch the
| video, your mileage may vary!)
|
| [1] https://ed-ac-uk.zoom.us/rec/share/_I-
| EeZA8_399ArdZ1GyKtM_rD... (link copied from the page, I hope it
| works)
| Joker_vD wrote:
| > But then you realize something genuinely weird: _There's
| nothing you can do to distinguish_ i _from -_ i.
|
| Relatedly, to this day I still don't know how distinguish a left-
| handed coordinate system from the right-handed one _purely
| algebraically_. Is the basis [(1,0,0), (0,1,0), (0,0,1)] left- or
| right-handed? I don 't know without a picture! Does anyone?
| solveit wrote:
| Stack the vectors up so it's a matrix and take the determinant.
| The sign tells you which one it is.
| gowld wrote:
| No, it does not. The determinant tells you whether two bases
| have the same or different handedness, not which one is
| "left" or "right".
| lanstin wrote:
| It's 2 cosets, one is arbitrarily left handed and the other
| arbitrarily right handed. If you are in an orientable space
| :) if not, then there's no global concept of left or right.
| a
| fredilo wrote:
| Formulated differently, you cannot determine left- and
| right-handedness but you can determine same-handedness.
| gowld wrote:
| No, it's impossible, even in principle, to answer that
| question. You can draw a picture for either answer.
|
| "Left" and "right" are a dipole. Neither one can exist without
| the other, and they are symmetric. It's the same issue as we
| have with the conjugates discussed in Galois Theory.
|
| In fact, in an algebraic (non-ordered/arithmetic/analytic)
| perspective, it's misleading to use the symbols + and - to
| label the conjugates in field extensions like sqrt2 and i. Left
| and Right are better names than + and - for those conjugate
| pairs. Only when we impose an arithmetic ordering (which is not
| needed in the theory of algebraic equalities) is it meaningful
| to use + and -: -sqrt(x) < 0 < +sqrt(x), where x is a positive
| real number.
|
| (and when x is a negative real number, we immediately see the
| problem with - again: -i and i are not separable via ordering
| with respect to 0.)
| kevinventullo wrote:
| Once you get to higher extensions where the roots are
| indistinguishable w.r.t. the base field, mathematicians will
| just write something like
|
| "Let a_1, ..., a_5 be the five roots of p(x)."
| fredilo wrote:
| The sentence you quote actually gives you the answer to your
| question but it is not completely obvious why. :)
|
| The complex numbers are essentially the theory of 2D-space. You
| are asking about 3D-space. The statement that you quoted tells
| you that you cannot distinguish between up and down in theory.
|
| Now, 2D-space is part of every 3D-space. There are multiple
| ways to see this. The easiest is to just drop the z-coordinate.
|
| Suppose you could distinguish left-handed and right-handed in
| 3D-space. In this case, you would have a way to distinguish up
| and down in the embedded 2D-space. However, you cannot do this
| distinction in 2D-space and therefore you cannot distinguish
| left-handed and right-handed in 3D-space.
| revskill wrote:
| The problem with many mathematics books, is it uses Math to teach
| Math !!!
|
| OK, it's fine in some cases, but it's like a gatekeeping itself,
| because in order to understand Math, you need to understand Math
| :)
| koolala wrote:
| Painting, Sculpture, Music, Art is a Language.
|
| We need the Toki Pona of Math. I hope Geometry one day becomes
| the foundation of Math again. Anyone can participate in
| Geometry just with a stick or VR headset.
| Koshkin wrote:
| I think everyone already knows some math before they start
| reading books on it.
| senderista wrote:
| Ian Stewart's book is excellent for self-study and has some
| fascinating historical background.
|
| https://www.taylorfrancis.com/books/mono/10.1201/97810032139...
| susam wrote:
| Galois Theory by Ian Stewart is an excellent book indeed! I've
| got a hard copy lying at home that I am currently reading
| slowly page by page. I am planning to host book club meetups
| with this book later this year, perhaps during the winter if I
| am able to make good progress with this book.
|
| In the meantime, if there is someone here who is interested in
| reading this type of books together and share updates with each
| other, I'd like to invite you to the IRC and Matrix channel
| named #bitwise [1][2] (the IRC and Matrix channels are bridged
| together, so you could join either one of them). The channel
| consists of some HN users as well as some users from other
| channels like ##math, ##physics, #cs, etc. It serves as an
| online space to share updates about mathematics and computation
| books you are reading and discuss their content.
|
| [1] https://web.libera.chat/#bitwise
|
| [2] https://app.element.io/#/room/#bitwise:matrix.org
| jonathanyc wrote:
| Thanks for this! I tried joining a few other channels on
| Libera like #robotics and ##typetheory but they're kinda
| dead. Usenet Newsgroups are also all spam. Will check it out.
| nyankosensei wrote:
| Another introduction from an historical point of view is
| "Galois Theory for Beginners: A Historical Perspective" by Jorg
| Bewersdorff
|
| https://bookstore.ams.org/view?ProductCode=STML/95
| jaymzcampbell wrote:
| I can agree with that, this is the primary textbook for the
| Open University's (excellent) Galois Theory MSc course
| (https://www.open.ac.uk/postgraduate/modules/m838. I really
| enjoyed my time on that course.
|
| I was also very interested in reading about the the original
| papers prior to the various advancements in mathematical
| thinking and notation that tend to reframe how the theory is
| taught today. For that I highly recommend Peter Neumanns "The
| mathematical writings of Evariste Galois"
| (https://ems.press/books/hem/102) - it has the french side by
| side with a direct English translation along with notes
| explaining the context and possible thought process (it also
| served as a fun way to read some more French whilst I was
| trying to learn the language).
| quibono wrote:
| I hope it's OK to hijack the thread a bit: could you please
| tell me whether you'd recommend the Open University? I think
| I might be interested in one of their Mathematics MSc. I
| don't know anyone personally who's gone so I'd love to hear
| what you thought of the experience please.
| jaymzcampbell wrote:
| Yes, I would certainly recommend them if you are personally
| motivated. I've previously studied at Imperial for my BEng
| and then did a Mathematics BSc and then followed up with an
| MSc with the OU and certainly as a mature student, I have
| loved my time with the OU.
|
| You get out of it very much what you put in. The materials
| on the undergrad side are excellent, at the MSc level you
| will basically be given a list of textbook resources, the
| odd OU prepared summary material and then a few online
| video tutorials (live) and then you get on with it.
|
| There are forums and you have direct contact with your
| tutor though speaking myself, I never really engaged with
| them. You'll have a few "TMAs" to do (tutor marked
| assignments) and that's where you'll get direct feedback on
| your approach and questions but you are encouraged to
| message tutors if you need some guidance, and when I have I
| have always had a good experience.
|
| My email is in my bio if you want to reach out, I'd be
| happy to send you some of the M838 course notes for you to
| get an idea of what you can expect.
| jorgenveisdal wrote:
| Love this!
| fredgrott wrote:
| Do not forget the numbers book covering history of numbers that
| Albert Einstein recommended....author is Tobias Dantzig...
| amai wrote:
| I guess you mean this one:
| https://en.wikipedia.org/wiki/Number:_The_Language_of_Scienc...
| enugu wrote:
| There is a nice topological proof which gives a more direct and
| visual understanding what solving by radicals means. It is quite
| short but might take some time to absorb the concepts.
|
| https://jfeldbrugge.github.io/Galois-Theory/
| artemonster wrote:
| I wonder, can you do an LLM in GF(2)?
| klyrs wrote:
| Somebody did a paper on GF(3)...
|
| https://arxiv.org/abs/2402.17764
| zengid wrote:
| ELI5 what Galois theory is?
| klyrs wrote:
| My kid's 8 and still doesn't know what polynomials are. This
| one's gonna be tough.
|
| lol @ the coward who downvoted me without chiming in with a
| 5yo-digestible treatise on Galois theory
| zengid wrote:
| how hard is it to say "an advanced theory about algebra",
| instead of downvoting?
| bubblyworld wrote:
| Sometimes when you move a shape around in front of you, you end
| up with the same shape. Maths people call this _symmetry_ , and
| have lots of names for different ways you can get back to the
| same shape.
|
| For instance, if you flip a square around you get back the same
| square. This is called _reflective_ symmetry.
|
| If you spin a triangle around you sometimes end up with the
| same triangle. This one is _rotational symmetry_.
|
| Galois spent a lot of time thinking about numbers instead of
| shapes. What he realised is that when you add and multiply
| numbers in lots of different ways, you sometimes end up with
| the same number at the end. And sometimes _different_ numbers,
| when added and multiplied in the _same_ way, _also_ give you
| the same number at the end.
|
| For instance, if you take 1, multiply it by itself and subtract
| 1, you get 1x1-1=0.
|
| If you do the same with -1, you get (-1)x(-1)-1=0. A different
| number, using the same pattern, gives us the same result.
|
| What we're seeing here is that there are some symmetries in
| _numbers_ , not just shapes! Galois theory is all about the
| nitty gritty of how these number symmetries work, how to find
| them, and how to use them to do interesting mathematics.
| zengid wrote:
| this is a beautiful explanation! Thank you!
| daitangio wrote:
| Btw, the life of Galois is quite interesting: he died very young,
| and was a quite clever mathematician...
| klyrs wrote:
| > ... and I hope you can list all of the groups of order < 8
| without having to think too hard.
|
| Early morning reaction: oh god I've forgotten all of my group
| theory, this is _bad_.
|
| After lunch: oh, right, there's only two composite numbers below
| 8.
| gowld wrote:
| Naming the groups of order 8 is harder than naming all the
| smaller groups.
| marshallward wrote:
| > I hope you can list all of the groups of order < 8 without
| having to think too hard.
|
| Welp, guess I'm out.
| Koshkin wrote:
| https://en.wikipedia.org/wiki/List_of_small_groups
| Koshkin wrote:
| _Galois Theory For Beginners_ by John Stillwell is the shortest
| introduction that I 've ever seen.
|
| https://www.scribd.com/document/81010821/GaloisTheoryForBegi...
| dmd wrote:
| https://chalkdustmagazine.com/blog/review-of-galois-knot-the...
| is probably the best review.
| gowld wrote:
| This is a joke not related to the submission or its topic.
| broabprobe wrote:
| Danny O'Brien's blog post A Touch of the Galois is my favorite
| writing on Galois,
|
| > Flunked two colleges, fought to restore the Republic,
| imprisoned in the Bastille, and managed to scribble down the
| thoughts that would lead to several major fields of mathematics,
| before dying in a duel -- either romantic or political -- at the
| age of twenty.
|
| https://www.oblomovka.com/wp/2012/09/11/touch-of-the-galois/
| bhasi wrote:
| The link appears to be down. Would love to read it though.
| Venkatesh10 wrote:
| The website is just plethora of knowledge and content in 90s
| design. Just pure bliss and I love it.
| andyayers wrote:
| There are a few interesting places where Galois Theory touches on
| compilation/programming.
|
| Abstract interpretation models a potentially infinite set of
| program behaviors onto a simpler (often finite) model that is
| (soundly) approximate and easier to reason about (via Galois
| connections); here the analogy is to Galois Theory connecting
| infinite fields with finite groups. I often think about this when
| working on Value Numbering for instance.
|
| Also (perhaps a bit of stretch) it's interesting to think of
| extending a computational domain (say integers) with additional
| values (say an error value) as a kind of field extension, and as
| with field extensions, sometimes (perhaps unexpectedly)
| complications arise (eg loss of unique factorization :: LLVM's
| poison & undef, or NaNs).
| Koshkin wrote:
| you must be writing lisp
| andyayers wrote:
| Not these days, but yes, years ago.
| SkiFire13 wrote:
| To extend on this, while abstract interpretation may sound a
| bit "abstract" (pun not intended), it is the basis for many
| techniques for software verification and compiler
| optimizations. At its core it basically allows you to soundly
| approximate the set of reachable states in a program, which in
| turn can be used to check that no "bad" state can be reached
| (for software verification) or that e.g. some checks are
| useless and can be removed (for compiler optimizations). Other
| applications include the new borrow checker for the Rust
| programming language, which is built on various dataflow passes
| to determine at each program point which variable is "live"
| and/or "borrowed".
| dboreham wrote:
| Although a simple EE, I learned Galois Theory in college
| (coincidentally also in Edinburgh, although <other-university>).
| In 4th/final/senior year there were various elective classes
| including Advanced Mathematics which I chose as a kind of
| masochistic challenge. The class was very small, and it turned
| out taught by a "real mathematician" who commuted from the
| Mathematics department every day. Even though I've had a great
| deal of mathematics education I think this was the only time the
| teacher was someone who did mathematics all-in (as in he created
| new mathematics, published papers etc.) as opposed to someone who
| had the job of teaching some field (sic) in mathematics.
|
| He taught Galois Theory using its application to coding theory
| for worked examples. That class was something of a turning point
| in my life to be honest. I'd never think of constructing a
| heptagon again, for example. Definitely avoided Duels, and
| Montparnasse. Ok joking aside, it caused the proverbial lightbulb
| to turn on in my brain, and helped tremendously in my career
| later when I ran into folks trying to seem smart because they
| understood ECC or ZKPs. It was like the extreme opposite of those
| people who say "I never used a single thing I learned in
| college".
| JadeNB wrote:
| > Ok joking aside, it caused the proverbial lightbulb to turn
| on in my brain, and helped tremendously in my career later when
| I ran into folks trying to seem smart because they understood
| ECC or ZKPs.
|
| Presumably it helped to know that there was something beyond
| these folks' knowledge, but did it help in any more direct way?
| dboreham wrote:
| The main benefit was to do with how I conceived mathematics
| as a subject:
|
| In the before times, it appeared to be an infinite linear
| journey with subjects already studied in the set of
| "understood" things, and everything else in the "hard" set of
| things. Proceeding on the journey, subjects are expected to
| become harder and harder until eventually you're defeated and
| have to stop trying to learn more mathematics.
|
| After that class, my conception of mathematics was of an
| extensively cross-linked tree of subjects where seemingly
| unrelated fields connect to each other, and where although
| the number of fields is large, it is not infinite, and with
| effort and time everything can be understood if required.
|
| To be fair, that was the stated objective of the guy teaching
| the class. On day 1, he said "my objective is to teach you
| enough that you can understand whatever mathematics you need
| to understand in the future" and darn it he pretty much
| succeeded.
|
| In summary, more of an attitude than specific knowledge of
| anything in particular.
| whitten wrote:
| What was the name of your professor ?
| dboreham wrote:
| Interesting question, and I don't remember. This was 40 years
| ago. I did find the final exam paper recently and marveled
| that at one time I was able to ace it. Notable in that there
| was a typo in one question that rendered it unsolvable. I was
| able to gain credit nevertheless by showing it to be
| unsolvable. I think that's the only case I remember where an
| exam had been printed with a major mistake.
| vladde wrote:
| That's an interesting way of holding a pen, never seen that
| before
|
| at 4:26 in https://ed-ac-
| uk.zoom.us/rec/play/qc1PCp8gTozfuRpMYKcTkPZQ2C...
| jmount wrote:
| What do people think about the Edwards Galois Theory book?
| will-burner wrote:
| Galois theory is the explanation and apex of theoretical math
| that you can motivate and talk about at a dinner table with
| people that don't even like math, lol.
|
| Start with the quadratic formula, everyone seems to have some
| recollection of this. Talk about solving for x in polynomials.
| Then discuss if you can always solve for x, and what does that
| even mean. If you graph a polynomial it crosses the x-axis so
| there's a solution for x, but does that mean you can solve for it
| in a formula (this alludes to the fundamental theorem of algebra
| that every polynomial of degree n has n solutions in the complex
| numbers)?
|
| It's tough to get the idea of solution by radicals and how that
| relates to what it means to have a formula for x in terms of the
| coefficients of the polynomial.
|
| Anyways, the punchline is that there's no formula for x using
| basic arithmetic operations up to taking radicals, where the
| formula is in terms of the coefficients of the polynomial for a
| general degree 5 or higher polynomial. Galois theory proves this.
|
| Galois is credited with this because it took a lot of imagination
| to think about how to formulate and prove that there is no
| formula. What does it mean to not have a formula? How do you
| formulate it properly and then prove it?
| gjm11 wrote:
| This isn't _quite_ right -- Abel proved that there 's no
| quintic formula before Galois came along. Galois theory gives a
| whole lot more insight, lets you understand why _some_ quintics
| do have solutions in radicals, etc., but Galois doesn 't (or at
| least shouldn't) get credited for proving that there isn't a
| quintic formula, because he wasn't the first to do that.
| will-burner wrote:
| Don't let the truth get in the way of a good story! hahahaha
|
| But yeah you're right
|
| edit: i don't recall Abel's proof, but Galois reformulation
| of what it means to be solvable by radicals, introducing the
| permutation group of the roots is the big thing in my mind.
| arjvik wrote:
| For a layman (I stopped short of Galois theory so far),
| what's different about the permutation groups of quintic
| roots and above that leads to this?
| will-burner wrote:
| In lay terms the best I can say is that for n greater
| than or equal to 5, the set of all possible permutations
| of n things is complicated. For n less than 5 the set of
| all possible permutations of n things is simple just
| because n is small. That's what leads to there being
| general formulas for n = 2, 3, 4.
|
| Galois translated whether a polynomial has a solution for
| x in terms of the coefficients using algebraic operations
| up to using radicals into a property of the group of
| permutations of the roots of the polynomial. The property
| of the group is whether the group is solvable. For n
| greater than or equal to 5, the general permutation group
| on n objects is not solvable but for n less than 5 is is.
| There just are not that many permutation groups for n =
| 2, 3, and 4 objects and all these permutation groups are
| solvable. Generically a group is not solvable and so we
| see this with larger n.
| javier_e06 wrote:
| I was put through the ringer on Louis Leithold "Calculus, with
| analytic, geometry". Heavy heavy book.
|
| "Do the exercises" teacher echoed over and over. I read the
| chapter, I followed the examples and proceed to the first problem
| in the unit.
|
| My answer was 64
|
| I go to the end of the book and the answer was 2 1/4
|
| I would try to reverse engineer the 2 and 1/4 to original
| problem... Nothing!
|
| I would ask a friend to the problem with me.. her answer was 16.
|
| Maybe divide by 8? that gets us 2, we are closer? Right. Why
| divide by 8? I don't know!
|
| Back in the there was no Internet or Kahn Academy. It was you and
| the red heavy book of Calculus with the desk lamp staring at you.
| Silently.
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