Posted by E-Reverance 1 day ago
Lots of things are torsors: position, currency values, calendar dates etc. the vales themselves are arbitrary, and translating/scaling them by some value doesn't make a functional difference. Torsors let us talk about these things without needing to make such an arbitrary choice a priori.
In the case of baseless logs, the underlying set is "information units", i.e. log 2 is bits, log e is nats, log 10 is digits, etc. The conversion factors give us the torsor's group, and picking a privileged unit is just a trivialization of the torsor.
The vector division notation is, similarly, encoding a g-torsor in precisely the same way as length units are.
The examples so far are all torsors with abelian groups, but specifying position both requires choosing an origin and a length unit. The group of this torsor is a suitable semidirect product between translation and scaling, which gives a non-abelian group.
Most of the time we just implicitly choose a trivialization, which often causes confusion because it identifies objects with operations on them, e.g. conflating vectors as positions with vectors as translations. The author's treatise on problems with geometric algebra [1] even brings up this point!
[0]:https://math.ucr.edu/home/baez/torsors.html
[1]:https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm...
Unfortunately, in mathematics there already is a long tradition of reusing common words to designate concepts that have no relationship whatsoever with the original meanings of those words. This obfuscates the content of many mathematical books or research papers, because even when they state trivial facts the statements are opaque for those unfamiliar with the specific jargon used in that niche branch of mathematics.
The hypothesis seems to be that the idea of affine spaces came out of that theory, for whatever reason, which was subsequently generalized to principle bundles and finally into what we have now. The point is that, at every step along the way, we want to connect the incrementally new ideas to existing ones, and creating a hard break with new, idiosyncratic terminology is itself obfuscatory.
My beef is more with use of the heavily-overloaded words "regular" and "normal" in math, which just seems like lazy naming:
> In the normal extension K/Q, every normal subgroup of the regular representation acts on a normal scheme that is regular in codimension one, whose normal bundle — orthonormal to the regular surface at each regular value — carries a normal operator whose spectrum follows a normal distribution over a space that is at once regular and normal, all indexed by a regular cardinal.
That's like 8 different meanings of normal and 6 different meanings of regular. lol
"a person who makes calculations, especially with a calculating machine."
Google ngram view:
https://books.google.com/ngrams/graph?content=computer&year_...
https://golem.ph.utexas.edu/category/2013/06/torsors_and_enr...
Consider in particular that use of ‘distance’
>I think you can look at adjoint profunctors from the unit category and show that they consist of giving a consistent ‘distance’ to every object, which in a torsor will be represented.
Regardless of the terminology, I thought it was interesting because I have never seen the logarithm thought about in that way.
Thanks for the writeup!
I wonder if we should really just call them... vectors? Like the thing that torsors do, being defined only relative to a choice of origin in some space / group, is exactly what displacement vectors do. So really they are just generalizations of the concept of a vector. (In this scheme I would be careful to _not_ refer to points as vectors, so as to reserve the term for things that act like, well, torsors. I happen to think that much pedagogical harm has been done by not distinguishing the two concepts, points and displacements, early on.)
Maybe. I think the term is just unfamiliar. The word "vector" is equally unhelpful when you first encounter it, but the concept has enough mindshare that we just acclimate.
Apparently, "vector" in the mathematical sense was coined by Hamilton in the mid 1800s, so around the same time as "torsor". It means something like "courier" which is sensible in Euclidean space but kind of divorced from the algebraic definition. You can still see the "carry along" roots if you squint, but I think the same is mostly true of "torsor", too.
In other words, instead of renaming things, maybe we should just evangelize more? Maybe the quintessential example should be radial directions just to hint at the historical terminology use?
https://www.google.com/books/edition/Trigonometry_for_Naviga...
See my other comment:
https://www.google.com/books/edition/Trigonometry_for_Naviga...
I found this book because I was a little rusty on my trig and most celestial navigation texts will just throw the PZX equation (and others) at you without breaking down what's actually being done with it on a mathematical level...it's just kind of treated like a magical black box without any discussion, and I'd rather have a complete understanding of what I'm doing and why. Having an application-specific approach also makes it a lot easier to learn.
I'm using it with Norie's Nautical Tables, which has the log tables and a whole lot else:
https://bluewaterweb.com/product/nories-nautical-tables-2025...
I'm sure there are plenty of free PDF's of log tables you can find though.
(I believe they used log tables on boats primarily because it's easier to use than a slide rule when everything is constantly rocking back and forth.)
Anyway, I was reading Merle B. Turner's Celestial for the Cruising Navigator, before I decided to focus on trig for a while. IMHO, it doesn't explain some of the trigonometric formulas as well as it should, or how they were derived, and I just can't learn that way. It's actually not a bad book, but I did find myself consulting a lot of outside resources (mainly with trig and some astronomy). The main problem with it to me is just not explaining how some formulas were derived. It's more dense than most books, but very informative.
For math, there's the book linked in the previous comment, and I'd also recommend looking at:
- An Introduction to Spherical Trigonometry by J.H. Clough-Smith -- I actually found Trigonometry for Navigating Officers in it's bibliography.
- The Elements of Navigation (1E) by Charles H. Cotter is an all-inclusive navigation book that starts with all the necessary trig in the first few chapters. A WORD OF WARNING THOUGH: I started reading the third edition of this book (revised by Lahiry) and it had so many mistakes in it I threw it out; honestly I'm not sure how it got published. Go on abebooks or biblio and buy the 1E if you're interested. I'm starting to work this book concurrently with Trigonometry for Navigating Officers and it's great so far, if dated (published 1958 IIRC).
For some books I own but haven't really started on in earnest yet,
- Celestial in the GPS Age by John Karl is supposedly an excellent resource, and I believe it even introduces a new method for fixing your position. It also gets into more of the "why".
- Dutton's Nautical Navigation (15E) by Thomas J. Cutler and Celestial Navigation: A Complete Home Study Course by David Burch are both supposed to be good as well as far telling you how to do everything, but don't appear to explain some of the "why", like the azimuth equation.
- A Short Guide to Celestial Navigation by Henning Umland is a great resource, but definitely leans more technical. You can find the PDF here:
https://www.celnav.de/page2.htm
Besides that, you'll obviously need a Nautical Almanac for the current year (you can find a PDF online), and you'll probably want a copy of:
- Bowditch (American Practical Navigator). This is the definitive reference for anything navigation, it's published by the USCG and you can also download it for free:
Part 1:
https://thenauticalalmanac.com/2024_Bowditch-_American_Pract...
Part 2:
https://thenauticalalmanac.com/2024_Bowditch-_American_Pract...
You can get a print version from Paradise Cay Publications, it's both parts of Bowditch in a single hardback:
https://www.paracay.com/2024-american-practical-navigator-bo...
Norie's isn't strictly necessary IMO unless you're doing some serious offshore boating like crossing to Hawai'i. Just use a calculator while you're learning. Learn to use Norie's after you're comfortable and keep it as a backup onboard.
You may also be interested in checking out Starpath, which is based in Seattle, and I believe you can even drop by and talk to somebody who's more knowledgeable than I:
They publish the David Burch books, and offer online courses.
Celestaire is another good shop for celnav:
Anyway, this was longer than I intended but I hope this puts you in the right direction!
I love book discussions and especially by someone like you (and Keller, who in the dedication to his Pascal book has: "to my father, who taught me the importance of learning, and to my mother, who taught me the importance of not doing it all the time."
Btw, to thank you, I'll share this hidden gem that will help making learning smooth and fun: "Learning & Memory" by W. Wickelgren-- I've seen dozens of books on this but only this walks the walk (the author practices cognitive psychology to make his book easy to learn from, and to remember).
Have a good day,
Eagga
https://www.starpath.com/catalog/books/1992.htm
This is a workbook and records the data used for navigation by sextant to Hawai'i. Good for practice.
It’s like audio where people say "dB" as if it answers the next question. Relative to what, measured how, and weighted for whom?
Author should brush up on https://en.wikipedia.org/wiki/Lie_theory
As developed in the article, informally, but somewhat sufficiently, the change of base formula shows that the choice of base is largely irrelevant: different bases give equivalent logarithms up to a constant factor.
The Taylor expansion of exp gives a more intrinsic and general definition of the exponential function. This allows exp to be generalised structurally to many algebraic settings, provided the relevant convergence conditions are met: for example, the complex exponential and its many possible logs, the matrix exponential, and so on…
Units are important as a sort-of type system, even at the conceptual level.
You are right that bases are not as important conceptually.
> The apparent magnitude of known objects can range from −26.832 for our Sun to about +31.5 for objects in deep space imaged by the Hubble Space Telescope.[3]
"Sound Power Level SWL", "Sound Pressure level SPL", and "Sound Intensity Level SIL" are different quantities which should not be confused. - https://sengpielaudio.com/calculator-soundpower.htm
A sound source produces sound power and this generates a sound pressure fluctuation in the air. Sound power is the distance independent cause of this, whereas sound pressure is the distance-dependent effect.
Sound pressure p is a "sound field quantity" and sound intensity I is a "sound energy quantity". In teachings these terms are not often separated sharply enough and sometimes are even set equal. But I ~ p2.
Understanding dB - http://www.jimprice.com/prosound/db.htm
dBFS - https://en.wikipedia.org/wiki/DBFS
Videos:
Understanding dB level by Paul McGowan - https://www.youtube.com/watch?v=t3Via4c8SlI
Paul explains 0dB and why there's a minus sign on volume - https://www.youtube.com/watch?v=NgEr6NEDPd4
The later reuse of “log” across valuations, dimension, vector fields, orders of vanishing is not so good. Those may be related ideas, but each needs a type signature: from what, to what, and preserving which operation?
So what do you do in practice? You have to normalize: you don't calculate log x, but instead log x/U for some scaling unit U. It's typical for U to be something like 1 mV or 1 W in electrical engineering, for example. This is completely legitimate, but it does mean that the thing that comes out needs a corresponding unit attached to it: dBmV, dBW, et cetera.
And it's really kind of important to be careful about that.
Charles Petzold's writings are always very clear and in-depth.
Sure we can, with some naive algebra. If we can take log(x,base) and drop the base, then we can also take pow(base,x) and drop the base. Since bits=log(2), then pow(bits)=2. You can probably connect it to the reverse of things, like integrals.
Also, for fun, I'll play with some notation tricks.
log(freq) = pitch
freq = pow(pitch)
octave = log(2)
400*Hz = 100*Hz*4 // the frequency 400 Hz equals 4 times 100 Hz
log(400*Hz) = log(100*Hz) + log(4)
log(400*Hz) = log(100*Hz) + 2*log(2)
log(400*Hz) = log(100*Hz) + 2*octave
log(400*Hz) = log(100*Hz) + 2*octave // the pitch of 400 Hz equals 2 octaves above the pitch of 100 Hz
cent = log(2)/1200
A4 = log(440*Hz)
B4 = A4 + 200*cent // the pitch B4 equals 200 cents above A4
B4 = log(440*Hz) + 200*log(2)/1200
B4 = log(440*Hz) + log(2^(2/12))
B4 = log(440*Hz * 2^(2/12))
pow(B4) = 493.883 Hz // the frequency of B4 equals 493.883 Hz
pow(log(440*Hz) + 200*log(2)/1200)
exp(ln(440) + 200*ln(2)/1200) dB_P = log(10)/10
dB_F = log(10)/20
log(10*V) = log(V) + 20*dB_F // the level of 10 V equals 20 dB more than the power level of 1 V.
SPL = 20*10^-6 * Pa
hearing_damage = log(SPL) + 90*dB_F // hearing damage occurs over 90 dB_F above SPL (neglecting A-weighting)
pow(hearing_damage) = pow(log(SPL) + 90*dB_F))
pow(hearing_damage) = pow(log(SPL) + 90*log(10)/20))
pow(hearing_damage) = SPL*pow(90*log(10)/20))
pow(hearing_damage) = SPL*31622.7766 // the pressure of hearing damage occurs above 31622 times SPL
pow(hearing_damage) = 0.632455532 Pa // the pressure of hearing damage occurs above 0.632 Pa
log(reference_unit) + value*dB_F (or dB_P)
log(reference_unit) - value*dB_F (or dB_P)
Nonetheless, where the author of TFA is correct is that logarithms are a single physical quantity, like length, area or volume, and that choosing the so called "base" is choosing the unit of measurement for logarithms.
Logarithms are included in the dimensional formulae of many derived physical quantities, e.g. for describing the attenuation or amplification of waves during their propagation, where one uses quantities like logarithm per length and logarithm per time.
Changing the "base" of logarithms modifies the numeric values of all derived physical quantities exactly in the same manner as changing any other fundamental unit of measurement, like the unit of length or the unit of time.
Like for any physical quantity, the complete value of a logarithm is independent of the unit of measurement, because it is the product between the numeric value and the unit of measurement. When the unit of measurement is changed, both the numeric value and the unit are changed and the product stays the same (i.e. the logarithm corresponds to the same ratio, regardless what base is used to compute a numeric value for the logarithm).
Nowadays, the unit of logarithms is normally chosen between the octave (binary logarithms), neper (hyperbolic logarithms) or bel (decimal logarithms).
The units of measurement for logarithms are not the bases, but the logarithms of the bases, which is why e.g. the value of the number "e", the base of the hyperbolic logarithms, is never needed in any computation. The only values that are needed are "ln 2" or its inverse "log2 e", which are used to convert the numeric values of logarithms when the unit of measurement is changed between those corresponding to binary logarithms and to hyperbolic logarithms (a.k.a. natural logarithms, but there is nothing more "natural" about hyperbolic logarithms than about any other kind of logarithms).
d(logₐx)/dx = 1/(x log(a))
As for changing the base changes the numbers, I have to wonder if you've done any advanced linear algebra or, more specifically, tensors. The whole point of a tensor is that it operates the same on an object regardless of the basis. Put another way, if a and b are two representations of the same object with different bases then T(a) and T(b) are equivalent if T(x) is a tensor.
My point is that any numbers are an arbitrary choice and they don't define the underlying structure. The author here is talking about logarithmic structure.
This btw is why you learn about different bases in linear algebra and converting between them. Or even polar coordinates vs cartesian coordinates (in high school, for some reason). They're priming you to learn about structure. You get to groups and learn that group A and B are isomorphic they have the same mathetmatical structure.
Even when the numbers change.
I use the word "logarithm" in its original sense, meaning "logarithmic quantity". Logarithms are a certain kind of quantity, which measures numeric ratios, like other quantities measure various things, e.g. plane angles, lengths, time or cardinal numbers, where the latter measure how many elements are in a set.
Even for cardinal numbers, where there is an obvious "natural" unit, the number "1", it is frequent in practice (e.g. when computing statistical quantities) to choose other units of measurement, like a thousand, a million, a billion, the Avogadro number, the Curie number, etc.
Both for a logarithm or for a cardinal number, like for a distance or an angle, the complete value is independent of the chosen unit of measurement, even if the numeric value changes.
As you say, while for a scalar quantity the complete value is independent of the unit of measurement, for a vector quantity or tensor quantity the complete value is also independent of the chosen reference system of coordinates, even if the numeric values of the components of a vector or tensor change when the reference system is changed.
However, all these have nothing to do with whether the term "baseless logarithm" makes sense.
You say that this should be used as a term with the meaning "logarithmic function" (because the family of functions defined by you is the same as the family of functions traditionally named "logarithmic functions", since Leonhard Euler).
I say that this claim is baseless itself, because the term "logarithmic function" has been in use for almost three centuries and there is absolutely no need to invent another term, which also does not make sense etymologically, because when computing any logarithmic function, i.e. any member of the function family that has the property mentioned by you, you need a concrete base value, i.e. no such function is baseless.
To post such a pattern allows the thought process to become distributed. Perhaps someone else will see the insight.
I don't mean that this is necessarily the case, but that it is where we are now: we have found ourself in a situation where we have way too many facts and not enough simple perspectives that make them useful and accessible.
Just my opinion, though.
It's the same as with software refactoring. If you refactor without a sense of what you want to get out of the refactor, how do you know whether you're refactoring the right things?
(Also just my opinion)
My clue-finding and pattern-matching and such is all based on philosophical aesthetics: something feels amiss when these patterns exist without being obvious; therefore they should be extracted and examined from various sides to see if a connection is found.