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!
Using the term "torsor" for that mathematical concept has been a very bad choice, both because the concept does not have any obvious relationship with the meaning of the word and because the word "torsor" had already been used for a very long time in classical mechanics for a very different concept, i.e. for the quantity that must be null for a rigid body to stay in equilibrium (i.e. the pair of a resultant force and a resultant torque).
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.
Words happen more than they are chosen, cf. "computer". The term "torsor" in this sense likely comes from the French "torseur" [0], which was used to describe rigid-body motions via a fundamental screw-like action.
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
>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.
I do know about torsors actually but I didn't think to link it from there. I guess I don't find the term very useful; it feels like things are still hard to think about even after you know it's a torsor!---but also, I think I need to get more familiar with the concept, because the other commenter on here who described my basis-logarithm as a "GL(V)-torsor" really said it much more succinctly than what I was hacking out manually.
Regardless of the terminology, I thought it was interesting because I have never seen the logarithm thought about in that way.
Thanks for the article. I do think your more elementary approach is good pedagogy since the subject is so broadly familiar already. I just like torsors, since they elegantly encode the "arbitrary choice" needed to deal with lots of objects.
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.)
> I wonder if we should really just call them... vectors?
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?
Meh. I'm fine with trying to rename things if it feels justified. And the more obscure the thing, the easier it is to rename in principle. Maybe the better name will help it be less obscure. But of course whether it works is proportional to how good of an argument there is for doing it, as it should be.
Logs are awesome. I started a math textbook from the 1920's a while ago, and all the calculations relied on tabulated logs, where you would convert the number to a log in a table to reduce the operation's degree, then convert back to the ordinary representation. This would reduce operations like finding cubed roots to division, would could be converted to log-log to be further reduced to subtraction before you would restore to ordinary notation. It feels like you're using a magic wormhole or something when you're doing this stuff by hand, it's really neat.
Another neat application, if a bit simplistic, are these mechanical paper computer that let you figure out your body-mass-index. They are basically two disks with logarithmic scales on them that you rotate relative to each other. Like a slide-rule, but circular. I think you can find them under the name 'BMI wheel'.
These exist for various medical usages. I've seen one to compute the due date of a pregnancy (from either the date of conception or the last missed period). This was at an obstetrician's office. It was probably dropped off by a sales rep and had the logo of some medication or other.
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:
Any other recommendations for getting into celestial navigation? I've used a sextant a few times and would like to purchase one but am aware that's only the hardware-side of things. Do the books you mentioned above provide sufficient tabulation for navigation? I sail in the Puget Sound for reference, thank you!
Well you can actually go an easier route, it just depends on whether you want to do it kind of mechanically or want to get intimate with all the workings of it. You can use HO 229/249 sight reduction tables and avoid a lot of fuss. Personally, I'd prefer to know how to calculate everything so I have a deeper understanding of what I'm doing. Plenty of books will walk you through all the steps, especially using the sight reduction tables, but I always walk away feeling like I'm missing something.
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:
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:
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:
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).
Anytime, and thank you for the rec, I just ordered it. I might also suggest to you "Make It Stick" (by Brown, Roediger, and McDaniel) if you are ever in want in another book on pedagogy.
Yep, we used manual math + some log tables for calculations in our school exams as late as last decade. Since calculators were not allowed. The exam would be such that you would need the log tables once or twice over the course of the exam. Example: dividing = lookup(a)-lookup(b) and then lookup that in the inverse log (i.e exp) tables.
Logarithms are laughably simple once you've fully internalized the meaning of the log function; it simply answers the question:
"To what power must I raise the base to get the argument?"
This is why the output tapers out as you increase the argument; because even if you increase the argument exponentially, you only need a fixed increment in the power to reach that number... So if you increase the argument only by a fixed amount (linearly) instead of exponentially, then it makes sense that the output will grow sub-linearly.
I remember when I was doing algebra with logs many years ago at school, I was applying rules to remove the log from one side of the equation.
Then when I got to uni, I had to revise the rules but it was kind of silly of me because those rules can be trivially derived if you just think about what the log function means. Turns out I had been solving equations with logs throughout school without understanding what they even meant... It's only at university that I actually bothered to learn them.
Actually TBH. I didn't even fully understand powers for some time even though I was doing calculus with them at school. I only fully understood powers once I properly internalized the concept of k-ary trees as a proxy.
It's one thing to be able to apply something, another to understand it. And I think to innovate with something, as a tool, it's not enough to be able to apply it. You must understand it.
Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of (perhaps) hereafter. But amongst all, none more profitable than this which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away from the work itself even the very numbers themselves that are to be multiplied, divided and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and subtraction, division by two or division by three.
This is what provides the intuition viz; convert multiplication/division/etc. of large numbers into addition/subtraction of two other smaller numbers. Logarithms as inverse of Exponentiation came much later. Starting with this generally confuses the student since they do not understand the point of it all.
Napier conceived the logarithm as the relationship between two particles moving along a line, one at constant speed and the other at a speed proportional to its distance from a fixed endpoint.
Since the speed is directly proportional to its remaining distance from the fixed endpoint, it therefore is a deceleration, which results in the characteristic "flattening" of the curve.
Further details for understanding the above can be found at
Priority, Parallel Discovery, and Pre-eminence: Napier, Burgi and the Early History of the Logarithm Relation (pdf) - http://www.numdam.org/item/RHM_2012__18_2_223_0.pdf
I find it surprising that logarithms and e (a.k.a. Napier’s constant), were developed and discovered only relatively recently in the history of mathematics despite how natural and fundamental they are.
The idea of exponential growth and the practice of charging interest in finance are both ancient. Surely an ancient mathematician would have investigated these in depth and discovered what Napier, Bernoulli and others found?
A while ago I was solving several infinite series of exponentials in the context of a problem concerning the half-life of medicines and I made frequent use of logarithms. That is when I started to wonder about their history.
Arithmetic series/Geometric series/Mathematical Tables (construction and lookup) were all well known in the ancient world. Wikipedia mentions Babylonians and Indians for logarithms (https://en.wikipedia.org/wiki/History_of_logarithms#Predeces...) but doubtless others too had variations of the same. Travel/Trade/Finance/Astronomy/Astrology would have been the main drivers before the modern scientific era.
We have lost a lot of knowledge accumulated and written down on biodegradable material (eg. papyri/palm leaves etc.) before the advent of the printing press made knowledge dissemination cheaper and easier. We then compounded the problem by dismissing everything before the beginning of our "scientific era" as being primitive/superstitious/non-methodical/etc. Only later on did we realize that many ancient civilizations were quite advanced in many aspects of mathematics and science though their way of approaching/inventing/recording was quite different from our "modern scientific method" and therefore we need to research them from a different pov and without condescension.
// The power to which I must raise 10 to get 100 is 2.
log10(100) = 2
// The power to which I must raise 10 to get 1000 is 3.
log10(1000) = 3
// The power to which I must raise 3 to get 27 is 3.
log3(27) = 3
Also it makes solving equations much more intuitive:
log3(x) = 4
^ This means; the power to which I must raise 3 to get x is 4.
So it follows logically that if I raise 3 to the power of 4, I will get x.
This makes it intuitive that this equation can be rewritten as:
x = 3 ^ 4
You don't even need to know the algebraic rule. I felt retarded when I figured this out.
This was a rule I had memorized before. It's even dumber and easier to infer than the rule to compute derivatives. I wonder why teachers even bother to teach you all these rules when they could just explain the fundamentals to you.
The above wikipedia page also details the properties, applications and generalization of the logarithm concept which are non-trivial.
As i pointed out above, that does not help in intuiting why it is helpful and needed. That is why you need to read the history of logarithms and see how we arrived at the above standard.
Napier actually calculated logarithms of sines for every minute from 0-90degrees to simplify astronomical calculations. The complexity/sizes involved, precision needed etc. can all be seen in this detailed paper walking you through the entire process of table construction; Napier’s ideal construction of the logarithms (pdf) - https://locomat.loria.fr/napier/napier1619construction.pdf
I think part of the confusion is that the base isn't always clear. A lot of times, when first exposed to logs, we are shown only base 10 or base e so it's just written as the word log or ln... This omission obscures the meaning and versatility of the function. Log and ln are essentially presented as separate constructs. Which is retarded.
Personally I only remember being briefly exposed to log and ln in school. I don't recall doing any complex algebra with them. So when I started doing algebra with them at university, I had to kind of re-learn from scratch.
I asked Claude about the weird way I was taught logs and it said this, which I found interesting:
There's a broader pattern here that shows up all over maths: notation and procedures get taught as the primary object, with the meaning treated as optional enrichment that "advanced" students might appreciate later. For a lot of people it's backwards — the meaning is the cheap, load-bearing thing, and the procedures are what should be derived from it. Logs are maybe the most egregant case because the gap between "incomprehensible button" and "obvious once stated" is a single plain-English sentence.
I had a weird relationship with Math growing up; I alternated between getting very high grades and terrible grades depending on the teacher. I didn't like all the notations and conventions of Math and the way it was taught, but I enjoyed it conceptually. It had ended badly in high school as I did poorly in advanced Math though I did quite well in all my other subjects so I got into a good Software Engineering degree at a top 50 university for engineering globally anyway.
But early in college, it occurred to me that I didn't understand Math concepts as intuitively as I understood programming concepts so I challenged myself to revisit everything from the beginning including numbers, addition, subtraction, fractions, roots, powers, probabilities, derivatives, integrals, vectors, matrices, calculus...
I had to free myself from thinking of Math as symbols on a piece of paper and think of it as being about actual quantities, transformations and combinations. I needed a completely new way to think about it and visualize every single step. When I was practicing calculus, I would stop at each step and try to visualize the equation. For example, when finding the 3D plane perpendicular to a point on a 3D curve, I would put effort into visualizing what happened to the equations across different dimensions at each step when I found the partial derivatives and combined them to get the 3D plane vectors.
My Math grades at university were quite good. I passed all the Math courses with ease and got several distinctions even.
The important properties of the logarithm are structural: we usually do not care about units or bases, except when carrying out an actual numerical computation.
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…
> The important properties of the logarithm are structural: we usually do not care about units or bases, except when carrying out an actual numerical computation.
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 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?
Or, to say a little more explicitly what you're getting at: when you take a logarithm of some quantity, log x, x absolutely must be unitless. There's no way whatsoever to take a logarithm of something with a unit attached. (This is an important and useful dimensional analysis check in formulas and long calculations!)
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.
Well, the brightness of celestial objects is also sometimes negative:
> 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]
And this is because Ptolemy’s catalog in which he ranked stars by their apparent brightness on a scale of one to six, one being the brightest. Ptolemy’s scale was (much later) retrofitted to a log scale (base 100^(1/5) or about 2.512), allowing extrapolation to both brighter and dimmer objects. The brightest of Ptolemy’s first-magnitude stars actually have negative magnitudes by the modern definition.
More specifically, 0 dB is the loudest sound the audio system is rated to produce without distortion. It's common to be able to actually drive systems harder than their specified engineering limits, which is why meters have a short positive dB section marked in red.
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.
I think what's going on with the complex logarithm is basically the same as the logarithm that outputs the set of all possible bases for a vector space. The complex logarithm produces a Z-torsor, and the basis logarithm produces a GL(V)-torsor. There's probably some way to represent a choice of branch cut as a part of the choice of the base of the complex logarithm, and similarly the choice of a specific basis as part of the choice of base of the vector space base logarithm.
Interesting, it did not occur to me of those as two instances of the same phenomenon. Although I still find the complex analytic one hard to think about.
The same idea comes up in physics. In quantum physics, the action S appears as the logarithm-like quantity behind the amplitude e^iS/(h^bar).
In statistical mechanics, entropy is the logarithm of the number of possible microstates Omega : S = log(Omega).
Although the concepts come from different parts of physics, they both reflect the same principle: using a log as a way to turn multiplicative relationships into additive ones.
>You might ask: if we have a baseless logarithm log(N), do we also have a “baseless exponential”?
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
I like the intuition that baseless logarithm notation gives, and it also avoids needing to choose a specific reference point. I can also directly calculate by choosing an arbitrary base:
Hah, I can use this to give decibels an actual unit.
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
Very helpful!! Imagine combining the goofy list of decibel suffixes into a uniform notation. Write the logarithm first so the + or - stays in the same spot.
The term "baseless logarithm" is really nonsensical and using it would be a great mistake.
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).
"Baseless logarithm" is not nonsencial. Given that:
d(logₐx)/dx = 1/(x log(a))
a baseless logarithm is simply a family of functions with similar properties. Perhaps it might be clearer if the author said something like the "logarithm property" rather than "baseless logarithm" but that's nit-picking and debatable.
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.
You use the word "logarithm" with the meaning "logarithmic function", i.e. a function whose argument is a ratio and whose result is a numeric value that gives the corresponding logarithm in a certain base.
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.
I read this kind of essay as a certain part of the arc by which new thoughts are formed: an act of large-scale pattern matching, laying out a bunch of cases which resemble each other, searching for the essential basis of the resemblance.
To post such a pattern allows the thought process to become distributed. Perhaps someone else will see the insight.
I happen to think that novel facts and theorems and proofs are way overrated. If you find a new fact it just goes into the giant pile of facts that are sitting around uselessly. The useful progress in math is comes from "refactoring" efforts to make things simpler and more intuitive.
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.
I agree that a lot of progress in math comes from refactorings and novel concepts that generalize neatly. My point is that those breakthroughs don't happen through refactoring in a vacuum, they happen because the refactoring is undertaken with a specific motivation. It often ends up having more general applications than the original motivation, but doing it without a specific motivation doesn't usually yield progress.
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?
I don't think that argument is actually true? You can refactor for purely aesthetic reasons and it may well turn out to help. Maybe you don't even know, going in, what the goal is; just that the way it works now isn't good and maybe if you start looking you'll find a better approach. Happens all the time, I think. Of course you can also refactor for aesthetic reasons and not help, but that's a skill issue.
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.
I think that's more about integrations/differentials not producing them (generally speaking). Physics likes to deal with integrals and differentiation as you calculate change over time or over spatial dimensions.
Eg. the integral of x^10 is x^11 / 11 + c. No hyper-operation appears and it's just another exponential (with a division).
The integral of log(x) is xlog(x) - x + c. So still basically just a logarithm
Even the integral of 2^x is just 2^x / log(2). Still basically the same thing.
There's no easy way to pull a hyper-operation out.
I'd say integrals or differentials are not as important on their own as the kinds of differential equations that come up in physics. Integrals and differentials don't produce hyperoperations from non-hyperoperations, but a solution to something as simple as y' - e^x y = 0 will have a double exponential.
However a lot of DEs in physics are linear second-order with coefficients that are most often constants or polynomials, and if they're not polynomial they are made to be so using series expansions, under reasonable assumptions. This already brings you a long way towards solving the problem. The resulting equations usually have trigonometric/exponential/special function solutions.
It's still possible that hyper-operations like a double exponential might come up in the study of some specific non-linear problems. As in the example above, if you have an exponential function as a coefficient in your differential equation you might get a double exponential in the solution somewhere. I'm not familiar with any specific physics examples though.
Wasn't there some scientific paper recently that proved that every operation can be represented as a logarithm? Like, the same as every logic gate can be derived from NAND gates
tl;dr: Being a homomorphism from a multiplicative structure into an additive structure isn't enough to grant it the logarithm title.
Although logarithms are certainly ubiquitous in mathematics, I don't think that the mappings that the article's author identifies as logarithms are appropriately viewed as such.
I can't endorse viewing dimension as a logarithm. It appears superficially logarithm-like because we typically (and somewhat unfortunately) write the direct sum of n copies of a vector space V as V^n rather than nV. Writing nV, we simply get the dimension identity dim(nV) = n dim(V). Writing nV instead of V^n also conveniently frees up V^n for the tensor product of n copies of V, with corresponding dimension identity dim(V^n) = dim(V)^n. So I don't think there's any "multiplicative-to-additive" business going on here at all.
Also, I don't think it's advisable to view the p-adic valuation ord_p as a logarithm, even though it's a homomorphisms from the multiplicative group of the rational or p-adic field into the additive group of the rational field. In fact, in many number theoretic contexts, the ratio log_p/ord_p is of particular interest.
I think a good rule of thumb for viewing a mapping as some kind of logarithm is that it has to have some relation with the Taylor expansion of log(1 + x) around x=0. Being a homomorphism from a multiplicative structure into an additive structure isn't enough to get the logarithm title.
Look, the whole thing actually makes sense and the core idea is pretty cool because it's true that a lot of stuff in math looks identical. But in my opinion this is way too much of a macro-level overgeneralization and you risk throwing everything into the same pot, which ends up diluting the actual point of things.I mean, if you take a hammer and a meat mallet, at the end of the day they're both chunks of metal used to hit stuff, but if you bunch them together without making any distinction, you lose track of why you use one to drive nails into a wall and the other to prep cutlets.Saying everything is just one big logarithm is a nice mental exercise, but I feel like it flattens out the differences too much and makes you lose the practical utility of the individual math tools, which are meant to solve completely different problems.
I'm a programmer so to me this brings to mind the idea of classes and subclasses. A program is implemented by having a set of classes. The classes can be organized into a class-hierarchy where they inherit methods from their ancestor-classes.
Now assume originally you did not have the feature of inheritance in your programming language so you would just create all the classes you need without orgnizing them into an inheritance-tree. Then you upgraded to a language that doe shave inheritance and you wanted to refactor your program to omit duplicate definitions of methods.
What kind of class-hierarchy would you come up with? There is no single way to do it. Some ways are better than others. There migh be more than one optimal way.
Same goes with generalization general, it is part of the language we create to describe things and there are many different languages we may come up with, some simpler, some more difficult to understand.
The baseless log here is just a torsor [0]!
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...
Using the term "torsor" for that mathematical concept has been a very bad choice, both because the concept does not have any obvious relationship with the meaning of the word and because the word "torsor" had already been used for a very long time in classical mechanics for a very different concept, i.e. for the quantity that must be null for a rigid body to stay in equilibrium (i.e. the pair of a resultant force and a resultant torque).
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.
Words happen more than they are chosen, cf. "computer". The term "torsor" in this sense likely comes from the French "torseur" [0], which was used to describe rigid-body motions via a fundamental screw-like action.
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
[0]:https://fr.wikipedia.org/wiki/Torseur
"computer" happened a while ago, it's usage predates the electronic computers as:
"a person who makes calculations, especially with a calculating machine."
Google ngram view:
https://books.google.com/ngrams/graph?content=computer&year_...
Yeah, see this thread --- I assume these guys haven't heard of the other meaning neither
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.
I do know about torsors actually but I didn't think to link it from there. I guess I don't find the term very useful; it feels like things are still hard to think about even after you know it's a torsor!---but also, I think I need to get more familiar with the concept, because the other commenter on here who described my basis-logarithm as a "GL(V)-torsor" really said it much more succinctly than what I was hacking out manually.
Regardless of the terminology, I thought it was interesting because I have never seen the logarithm thought about in that way.
Thanks for the article. I do think your more elementary approach is good pedagogy since the subject is so broadly familiar already. I just like torsors, since they elegantly encode the "arbitrary choice" needed to deal with lots of objects.
Thanks for the writeup!
glad you liked it
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.)
> I wonder if we should really just call them... vectors?
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?
Meh. I'm fine with trying to rename things if it feels justified. And the more obscure the thing, the easier it is to rename in principle. Maybe the better name will help it be less obscure. But of course whether it works is proportional to how good of an argument there is for doing it, as it should be.
Thanks for sharing, very interesting. I wonder how this maps to swe
Logs are awesome. I started a math textbook from the 1920's a while ago, and all the calculations relied on tabulated logs, where you would convert the number to a log in a table to reduce the operation's degree, then convert back to the ordinary representation. This would reduce operations like finding cubed roots to division, would could be converted to log-log to be further reduced to subtraction before you would restore to ordinary notation. It feels like you're using a magic wormhole or something when you're doing this stuff by hand, it's really neat.
The physical version of that magic wormhole is called a slide rule.
Another neat application, if a bit simplistic, are these mechanical paper computer that let you figure out your body-mass-index. They are basically two disks with logarithmic scales on them that you rotate relative to each other. Like a slide-rule, but circular. I think you can find them under the name 'BMI wheel'.
These exist for various medical usages. I've seen one to compute the due date of a pregnancy (from either the date of conception or the last missed period). This was at an obstetrician's office. It was probably dropped off by a sales rep and had the logo of some medication or other.
Indeed, you can get yours today https://www.instructables.com/Make-a-simple-paper-slide-rule...
Got a PDF? I love old books like this.
Here you go:
https://www.google.com/books/edition/Trigonometry_for_Naviga...
See my other comment:
https://news.ycombinator.com/item?id=48623646
care to share the name of the said book?
Trigonometry for Navigating Officers by WP Winter
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.)
Any other recommendations for getting into celestial navigation? I've used a sextant a few times and would like to purchase one but am aware that's only the hardware-side of things. Do the books you mentioned above provide sufficient tabulation for navigation? I sail in the Puget Sound for reference, thank you!
Well you can actually go an easier route, it just depends on whether you want to do it kind of mechanically or want to get intimate with all the workings of it. You can use HO 229/249 sight reduction tables and avoid a lot of fuss. Personally, I'd prefer to know how to calculate everything so I have a deeper understanding of what I'm doing. Plenty of books will walk you through all the steps, especially using the sight reduction tables, but I always walk away feeling like I'm missing something.
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:
https://starpath.com/
They publish the David Burch books, and offer online courses.
Celestaire is another good shop for celnav:
https://www.celestaire.com/
Anyway, this was longer than I intended but I hope this puts you in the right direction!
All that being said, honestly I'd just review trigonometry a little and pick up a copy of Celestial in the GPS Age and fill in the gaps from there.
Thank you very much!
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
Anytime, and thank you for the rec, I just ordered it. I might also suggest to you "Make It Stick" (by Brown, Roediger, and McDaniel) if you are ever in want in another book on pedagogy.
Appreciate it :)
Sorry, last thing:
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.
Yep, we used manual math + some log tables for calculations in our school exams as late as last decade. Since calculators were not allowed. The exam would be such that you would need the log tables once or twice over the course of the exam. Example: dividing = lookup(a)-lookup(b) and then lookup that in the inverse log (i.e exp) tables.
That's a lot of ways to think about logarithms.
Logarithms are laughably simple once you've fully internalized the meaning of the log function; it simply answers the question:
"To what power must I raise the base to get the argument?"
This is why the output tapers out as you increase the argument; because even if you increase the argument exponentially, you only need a fixed increment in the power to reach that number... So if you increase the argument only by a fixed amount (linearly) instead of exponentially, then it makes sense that the output will grow sub-linearly.
I remember when I was doing algebra with logs many years ago at school, I was applying rules to remove the log from one side of the equation.
Then when I got to uni, I had to revise the rules but it was kind of silly of me because those rules can be trivially derived if you just think about what the log function means. Turns out I had been solving equations with logs throughout school without understanding what they even meant... It's only at university that I actually bothered to learn them.
Actually TBH. I didn't even fully understand powers for some time even though I was doing calculus with them at school. I only fully understood powers once I properly internalized the concept of k-ary trees as a proxy.
It's one thing to be able to apply something, another to understand it. And I think to innovate with something, as a tool, it's not enough to be able to apply it. You must understand it.
A better way to understand logarithms is to start with the original motivation from Napier himself (https://sites.pitt.edu/~super1/lecture/lec44911/005.htm);
Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of (perhaps) hereafter. But amongst all, none more profitable than this which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away from the work itself even the very numbers themselves that are to be multiplied, divided and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and subtraction, division by two or division by three.
This is what provides the intuition viz; convert multiplication/division/etc. of large numbers into addition/subtraction of two other smaller numbers. Logarithms as inverse of Exponentiation came much later. Starting with this generally confuses the student since they do not understand the point of it all.
From https://en.wikipedia.org/wiki/History_of_logarithms;
Napier conceived the logarithm as the relationship between two particles moving along a line, one at constant speed and the other at a speed proportional to its distance from a fixed endpoint.
Since the speed is directly proportional to its remaining distance from the fixed endpoint, it therefore is a deceleration, which results in the characteristic "flattening" of the curve.
Further details for understanding the above can be found at Priority, Parallel Discovery, and Pre-eminence: Napier, Burgi and the Early History of the Logarithm Relation (pdf) - http://www.numdam.org/item/RHM_2012__18_2_223_0.pdf
I find it surprising that logarithms and e (a.k.a. Napier’s constant), were developed and discovered only relatively recently in the history of mathematics despite how natural and fundamental they are.
The idea of exponential growth and the practice of charging interest in finance are both ancient. Surely an ancient mathematician would have investigated these in depth and discovered what Napier, Bernoulli and others found?
A while ago I was solving several infinite series of exponentials in the context of a problem concerning the half-life of medicines and I made frequent use of logarithms. That is when I started to wonder about their history.
Arithmetic series/Geometric series/Mathematical Tables (construction and lookup) were all well known in the ancient world. Wikipedia mentions Babylonians and Indians for logarithms (https://en.wikipedia.org/wiki/History_of_logarithms#Predeces...) but doubtless others too had variations of the same. Travel/Trade/Finance/Astronomy/Astrology would have been the main drivers before the modern scientific era.
Here is an interesting book; The History of Mathematical Tables: from Sumer to Spreadsheets - https://en.wikipedia.org/wiki/The_History_of_Mathematical_Ta...
We have lost a lot of knowledge accumulated and written down on biodegradable material (eg. papyri/palm leaves etc.) before the advent of the printing press made knowledge dissemination cheaper and easier. We then compounded the problem by dismissing everything before the beginning of our "scientific era" as being primitive/superstitious/non-methodical/etc. Only later on did we realize that many ancient civilizations were quite advanced in many aspects of mathematics and science though their way of approaching/inventing/recording was quite different from our "modern scientific method" and therefore we need to research them from a different pov and without condescension.
PS: History of Mathematics - https://en.wikipedia.org/wiki/History_of_mathematics
I find my explanation simpler.
// The power to which I must raise 10 to get 100 is 2.
log10(100) = 2
// The power to which I must raise 10 to get 1000 is 3.
log10(1000) = 3
// The power to which I must raise 3 to get 27 is 3.
log3(27) = 3
Also it makes solving equations much more intuitive:
log3(x) = 4
^ This means; the power to which I must raise 3 to get x is 4. So it follows logically that if I raise 3 to the power of 4, I will get x. This makes it intuitive that this equation can be rewritten as:
x = 3 ^ 4
You don't even need to know the algebraic rule. I felt retarded when I figured this out. This was a rule I had memorized before. It's even dumber and easier to infer than the rule to compute derivatives. I wonder why teachers even bother to teach you all these rules when they could just explain the fundamentals to you.
That is just the definition of Logarithm which is what is taught to all students today i.e.
Given a^x = b we define log_a(b) = x where 'a' is a +ve real number - https://en.wikipedia.org/wiki/Logarithm#Definition
The above wikipedia page also details the properties, applications and generalization of the logarithm concept which are non-trivial.
As i pointed out above, that does not help in intuiting why it is helpful and needed. That is why you need to read the history of logarithms and see how we arrived at the above standard.
Napier actually calculated logarithms of sines for every minute from 0-90degrees to simplify astronomical calculations. The complexity/sizes involved, precision needed etc. can all be seen in this detailed paper walking you through the entire process of table construction; Napier’s ideal construction of the logarithms (pdf) - https://locomat.loria.fr/napier/napier1619construction.pdf
I think part of the confusion is that the base isn't always clear. A lot of times, when first exposed to logs, we are shown only base 10 or base e so it's just written as the word log or ln... This omission obscures the meaning and versatility of the function. Log and ln are essentially presented as separate constructs. Which is retarded.
Personally I only remember being briefly exposed to log and ln in school. I don't recall doing any complex algebra with them. So when I started doing algebra with them at university, I had to kind of re-learn from scratch.
I asked Claude about the weird way I was taught logs and it said this, which I found interesting:
There's a broader pattern here that shows up all over maths: notation and procedures get taught as the primary object, with the meaning treated as optional enrichment that "advanced" students might appreciate later. For a lot of people it's backwards — the meaning is the cheap, load-bearing thing, and the procedures are what should be derived from it. Logs are maybe the most egregant case because the gap between "incomprehensible button" and "obvious once stated" is a single plain-English sentence.
What made you want to understand it or did it happen upon you in college
It happened during college.
I had a weird relationship with Math growing up; I alternated between getting very high grades and terrible grades depending on the teacher. I didn't like all the notations and conventions of Math and the way it was taught, but I enjoyed it conceptually. It had ended badly in high school as I did poorly in advanced Math though I did quite well in all my other subjects so I got into a good Software Engineering degree at a top 50 university for engineering globally anyway.
But early in college, it occurred to me that I didn't understand Math concepts as intuitively as I understood programming concepts so I challenged myself to revisit everything from the beginning including numbers, addition, subtraction, fractions, roots, powers, probabilities, derivatives, integrals, vectors, matrices, calculus...
I had to free myself from thinking of Math as symbols on a piece of paper and think of it as being about actual quantities, transformations and combinations. I needed a completely new way to think about it and visualize every single step. When I was practicing calculus, I would stop at each step and try to visualize the equation. For example, when finding the 3D plane perpendicular to a point on a 3D curve, I would put effort into visualizing what happened to the equations across different dimensions at each step when I found the partial derivatives and combined them to get the 3D plane vectors.
My Math grades at university were quite good. I passed all the Math courses with ease and got several distinctions even.
This essay needs a type system. Every time it says “log” it should say: log of what, into what?
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
The important properties of the logarithm are structural: we usually do not care about units or bases, except when carrying out an actual numerical computation.
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…
> The important properties of the logarithm are structural: we usually do not care about units or bases, except when carrying out an actual numerical computation.
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 first section details how the author thinks of "log N" with no base as an abstract object rather than a number. Or what are you referring to?
The first section is the good part.
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?
Or, to say a little more explicitly what you're getting at: when you take a logarithm of some quantity, log x, x absolutely must be unitless. There's no way whatsoever to take a logarithm of something with a unit attached. (This is an important and useful dimensional analysis check in formulas and long calculations!)
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.
I still don't understand why audio dB are negative. That's relative to what? What happens at 0dB?
Well, the brightness of celestial objects is also sometimes negative:
> 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]
See https://en.wikipedia.org/wiki/Apparent_magnitude
And this is because Ptolemy’s catalog in which he ranked stars by their apparent brightness on a scale of one to six, one being the brightest. Ptolemy’s scale was (much later) retrofitted to a log scale (base 100^(1/5) or about 2.512), allowing extrapolation to both brighter and dimmer objects. The brightest of Ptolemy’s first-magnitude stars actually have negative magnitudes by the modern definition.
0db is usually defined as the loudest sound that the audio system can produce. Hence, everything else must be negative.
More specifically, 0 dB is the loudest sound the audio system is rated to produce without distortion. It's common to be able to actually drive systems harder than their specified engineering limits, which is why meters have a short positive dB section marked in red.
Of course, typical of the wonderful ambiguity of decibels, 0 dB is also usually defined as the quietest sound that the human ear can perceive.
https://en.wikipedia.org/wiki/Absolute_threshold_of_hearing
That's why important to give the scale. dBfs is full scale level, and db SPL is sound pressure level.
Yep.
"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.
That is dB full scale where 0 is an absolute ceiling and you can deduct from there.
Articles:
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
See also https://news.ycombinator.com/item?id=48632331
I think what's going on with the complex logarithm is basically the same as the logarithm that outputs the set of all possible bases for a vector space. The complex logarithm produces a Z-torsor, and the basis logarithm produces a GL(V)-torsor. There's probably some way to represent a choice of branch cut as a part of the choice of the base of the complex logarithm, and similarly the choice of a specific basis as part of the choice of base of the vector space base logarithm.
Interesting, it did not occur to me of those as two instances of the same phenomenon. Although I still find the complex analytic one hard to think about.
Charles Petzold's The Lost Art of Logarithms is a great read (still a work in progress).
https://www.lostartoflogarithms.com/
This looks great; thanks for the pointer.
Charles Petzold's writings are always very clear and in-depth.
The same idea comes up in physics. In quantum physics, the action S appears as the logarithm-like quantity behind the amplitude e^iS/(h^bar). In statistical mechanics, entropy is the logarithm of the number of possible microstates Omega : S = log(Omega). Although the concepts come from different parts of physics, they both reflect the same principle: using a log as a way to turn multiplicative relationships into additive ones.
>You might ask: if we have a baseless logarithm log(N), do we also have a “baseless exponential”?
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.
I like the intuition that baseless logarithm notation gives, and it also avoids needing to choose a specific reference point. I can also directly calculate by choosing an arbitrary base:Hah, I can use this to give decibels an actual unit.
Very helpful!! Imagine combining the goofy list of decibel suffixes into a uniform notation. Write the logarithm first so the + or - stays in the same spot. https://en.wikipedia.org/wiki/Decibel#List_of_suffixesTrue, I guess you can just 'curry' exponentiation and say that's a baseless power. I couldn't find a clean notation for it so I gave up..
The term "baseless logarithm" is really nonsensical and using it would be a great mistake.
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).
"Baseless logarithm" is not nonsencial. Given that:
a baseless logarithm is simply a family of functions with similar properties. Perhaps it might be clearer if the author said something like the "logarithm property" rather than "baseless logarithm" but that's nit-picking and debatable.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.
You use the word "logarithm" with the meaning "logarithmic function", i.e. a function whose argument is a ratio and whose result is a numeric value that gives the corresponding logarithm in a certain base.
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.
I can't believe he called normal logarithms 'based'
All this would be way more interesting if it actually helped to demonstrate a novel mathematical fact. Right now it's more like notational play.
I read this kind of essay as a certain part of the arc by which new thoughts are formed: an act of large-scale pattern matching, laying out a bunch of cases which resemble each other, searching for the essential basis of the resemblance.
To post such a pattern allows the thought process to become distributed. Perhaps someone else will see the insight.
I happen to think that novel facts and theorems and proofs are way overrated. If you find a new fact it just goes into the giant pile of facts that are sitting around uselessly. The useful progress in math is comes from "refactoring" efforts to make things simpler and more intuitive.
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.
I agree that a lot of progress in math comes from refactorings and novel concepts that generalize neatly. My point is that those breakthroughs don't happen through refactoring in a vacuum, they happen because the refactoring is undertaken with a specific motivation. It often ends up having more general applications than the original motivation, but doing it without a specific motivation doesn't usually yield progress.
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)
I don't think that argument is actually true? You can refactor for purely aesthetic reasons and it may well turn out to help. Maybe you don't even know, going in, what the goal is; just that the way it works now isn't good and maybe if you start looking you'll find a better approach. Happens all the time, I think. Of course you can also refactor for aesthetic reasons and not help, but that's a skill issue.
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.
Does this answer the question of why we see hyperoperations until exponentiation in physics, but not higher?
I think that's more about integrations/differentials not producing them (generally speaking). Physics likes to deal with integrals and differentiation as you calculate change over time or over spatial dimensions.
Eg. the integral of x^10 is x^11 / 11 + c. No hyper-operation appears and it's just another exponential (with a division).
The integral of log(x) is xlog(x) - x + c. So still basically just a logarithm
Even the integral of 2^x is just 2^x / log(2). Still basically the same thing.
There's no easy way to pull a hyper-operation out.
I'd say integrals or differentials are not as important on their own as the kinds of differential equations that come up in physics. Integrals and differentials don't produce hyperoperations from non-hyperoperations, but a solution to something as simple as y' - e^x y = 0 will have a double exponential.
However a lot of DEs in physics are linear second-order with coefficients that are most often constants or polynomials, and if they're not polynomial they are made to be so using series expansions, under reasonable assumptions. This already brings you a long way towards solving the problem. The resulting equations usually have trigonometric/exponential/special function solutions.
It's still possible that hyper-operations like a double exponential might come up in the study of some specific non-linear problems. As in the example above, if you have an exponential function as a coefficient in your differential equation you might get a double exponential in the solution somewhere. I'm not familiar with any specific physics examples though.
Wasn't there some scientific paper recently that proved that every operation can be represented as a logarithm? Like, the same as every logic gate can be derived from NAND gates
Was it this exp-minus-log arxiv paper?: https://arxiv.org/html/2603.21852v2
This sentiment (not necessarily the content) is what I'm striving to communicate with Mag World[0] (website and podcast so far).
[0] magworld.pw
tl;dr: Being a homomorphism from a multiplicative structure into an additive structure isn't enough to grant it the logarithm title.
Although logarithms are certainly ubiquitous in mathematics, I don't think that the mappings that the article's author identifies as logarithms are appropriately viewed as such.
I can't endorse viewing dimension as a logarithm. It appears superficially logarithm-like because we typically (and somewhat unfortunately) write the direct sum of n copies of a vector space V as V^n rather than nV. Writing nV, we simply get the dimension identity dim(nV) = n dim(V). Writing nV instead of V^n also conveniently frees up V^n for the tensor product of n copies of V, with corresponding dimension identity dim(V^n) = dim(V)^n. So I don't think there's any "multiplicative-to-additive" business going on here at all.
Also, I don't think it's advisable to view the p-adic valuation ord_p as a logarithm, even though it's a homomorphisms from the multiplicative group of the rational or p-adic field into the additive group of the rational field. In fact, in many number theoretic contexts, the ratio log_p/ord_p is of particular interest.
I think a good rule of thumb for viewing a mapping as some kind of logarithm is that it has to have some relation with the Taylor expansion of log(1 + x) around x=0. Being a homomorphism from a multiplicative structure into an additive structure isn't enough to get the logarithm title.
IIRC, Knuth use lg for logarithm base 2.
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Look, the whole thing actually makes sense and the core idea is pretty cool because it's true that a lot of stuff in math looks identical. But in my opinion this is way too much of a macro-level overgeneralization and you risk throwing everything into the same pot, which ends up diluting the actual point of things.I mean, if you take a hammer and a meat mallet, at the end of the day they're both chunks of metal used to hit stuff, but if you bunch them together without making any distinction, you lose track of why you use one to drive nails into a wall and the other to prep cutlets.Saying everything is just one big logarithm is a nice mental exercise, but I feel like it flattens out the differences too much and makes you lose the practical utility of the individual math tools, which are meant to solve completely different problems.
I'm a programmer so to me this brings to mind the idea of classes and subclasses. A program is implemented by having a set of classes. The classes can be organized into a class-hierarchy where they inherit methods from their ancestor-classes.
Now assume originally you did not have the feature of inheritance in your programming language so you would just create all the classes you need without orgnizing them into an inheritance-tree. Then you upgraded to a language that doe shave inheritance and you wanted to refactor your program to omit duplicate definitions of methods.
What kind of class-hierarchy would you come up with? There is no single way to do it. Some ways are better than others. There migh be more than one optimal way.
Same goes with generalization general, it is part of the language we create to describe things and there are many different languages we may come up with, some simpler, some more difficult to understand.
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