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.)
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.)
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)
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.
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.
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.
To post such a pattern allows the thought process to become distributed. Perhaps someone else will see the insight.
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.
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.
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.
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
// 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.
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 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.
[0] magworld.pw
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.