9 ms·

Everything is logarithms

316 points by E-Reverance 3mo ago

yaccb3 3mo ago
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.
- galaxyLogic 3mo ago
  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.
badlibrarian 3mo ago
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 https://en.wikipedia.org/wiki/Lie_theory
- jmyeet 3mo ago
  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?
  - badlibrarian 3mo ago
    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?
    - exmadscientist 3mo ago
      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.
- rq1 3mo ago
  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…
jongjong 3mo ago
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.
- whattheheckheck 3mo ago
  What made you want to understand it or did it happen upon you in college
  - jongjong 3mo ago
    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.
amelius 3mo ago
Does this answer the question of why we see hyperoperations until exponentiation in physics, but not higher?
- AnotherGoodName 3mo ago
  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.
  - renyicircle 3mo ago
    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.
helterskelter 3mo ago
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.
- badlibrarian 3mo ago
  The physical version of that magic wormhole is called a slide rule.
  - eru 3mo ago
    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'.
    - madcaptenor 3mo ago
      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.
      - eru 3mo ago
        Yes. If you are clever enough, you can encode a lot of formulas in geometry and paper. Navigators of yore used to have a lot of these tools.
  - utopiah 3mo ago
saulpw 3mo ago
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
aesthesia 3mo ago
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.
- ajkjk 3mo ago
  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.
kfse 3mo ago
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.
- sixo 3mo ago
  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.
- ajkjk 3mo ago
  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.
  - kfse 3mo ago
    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)
    - ajkjk 3mo ago
      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.
anArbitraryOne 3mo ago
I can't believe he called normal logarithms 'based'
xelxebar 3mo ago
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 https://math.ucr.edu/home/baez/torsors.html [1]:https://alexkritchevsky.com/2024/02/28/geometric-algebra.html https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm...
- ajkjk 3mo ago
  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.
  - xelxebar 3mo ago
    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!
SadErn 3mo ago
[dead]
monkamonme 3mo ago
[flagged]
adrian_b 3mo ago
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).
- jmyeet 3mo ago
  "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. Even when the numbers change.
myzek 3mo ago
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
- ebolyen 3mo ago
  Was it this exp-minus-log arxiv paper?: https://arxiv.org/html/2603.21852v2 https://arxiv.org/html/2603.21852v2
psychoslave 3mo ago
IIRC, Knuth use lg for logarithm base 2.
orc00 3mo ago
Charles Petzold's The Lost Art of Logarithms is a great read (still a work in progress). https://www.lostartoflogarithms.com/ https://www.lostartoflogarithms.com/
- rramadass 3mo ago
  This looks great; thanks for the pointer. Charles Petzold's writings are always very clear and in-depth.
amavect 3mo ago
>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: pow(log(440*Hz) + 200*log(2)/1200) exp(ln(440) + 200*ln(2)/1200)
- ajkjk 3mo ago
  True, 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..
- amavect 3mo ago
  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. log(reference_unit) + value*dB_F (or dB_P) log(reference_unit) - value*dB_F (or dB_P) https://en.wikipedia.org/wiki/Decibel#List_of_suffixes https://en.wikipedia.org/wiki/Decibel#List_of_suffixes
GL26 3mo ago
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.
math_dandy 3mo ago
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.

316 points by E-Reverance 3mo ago

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.

galaxyLogic 3mo ago
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.

badlibrarian 3mo ago

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 https://en.wikipedia.org/wiki/Lie_theory

jmyeet 3mo ago
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?
- badlibrarian 3mo ago
  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?
  - exmadscientist 3mo ago
    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.
rq1 3mo ago
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…

jongjong 3mo ago

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.

whattheheckheck 3mo ago
What made you want to understand it or did it happen upon you in college
- jongjong 3mo ago
  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.

amelius 3mo ago

Does this answer the question of why we see hyperoperations until exponentiation in physics, but not higher?

AnotherGoodName 3mo ago
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.
- renyicircle 3mo ago
  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.

helterskelter 3mo ago

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.

badlibrarian 3mo ago
The physical version of that magic wormhole is called a slide rule.
- eru 3mo ago
  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'.
  - madcaptenor 3mo ago
    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.
    - eru 3mo ago
      Yes. If you are clever enough, you can encode a lot of formulas in geometry and paper. Navigators of yore used to have a lot of these tools.
- utopiah 3mo ago

saulpw 3mo ago

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

aesthesia 3mo ago

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.

ajkjk 3mo ago
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.

kfse 3mo ago

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.

sixo 3mo ago
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.
ajkjk 3mo ago
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.
- kfse 3mo ago
  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)
  - ajkjk 3mo ago
    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.

anArbitraryOne 3mo ago

I can't believe he called normal logarithms 'based'

xelxebar 3mo ago

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 https://math.ucr.edu/home/baez/torsors.html [1]:https://alexkritchevsky.com/2024/02/28/geometric-algebra.html https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm...

ajkjk 3mo ago
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.
- xelxebar 3mo ago
  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!

SadErn 3mo ago

[dead]

monkamonme 3mo ago

[flagged]

adrian_b 3mo ago

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).

jmyeet 3mo ago
"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. Even when the numbers change.

myzek 3mo ago

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

ebolyen 3mo ago
Was it this exp-minus-log arxiv paper?: https://arxiv.org/html/2603.21852v2 https://arxiv.org/html/2603.21852v2

psychoslave 3mo ago

IIRC, Knuth use lg for logarithm base 2.

orc00 3mo ago

Charles Petzold's The Lost Art of Logarithms is a great read (still a work in progress). https://www.lostartoflogarithms.com/ https://www.lostartoflogarithms.com/

rramadass 3mo ago
This looks great; thanks for the pointer. Charles Petzold's writings are always very clear and in-depth.

amavect 3mo ago

>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: pow(log(440*Hz) + 200*log(2)/1200) exp(ln(440) + 200*ln(2)/1200)

ajkjk 3mo ago
True, 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..
amavect 3mo ago
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. log(reference_unit) + value*dB_F (or dB_P) log(reference_unit) - value*dB_F (or dB_P) https://en.wikipedia.org/wiki/Decibel#List_of_suffixes https://en.wikipedia.org/wiki/Decibel#List_of_suffixes

GL26 3mo ago

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.

math_dandy 3mo ago

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.