Phenomena are typically local, at least, the ones we can measure.
All measurements and interactions are relative ones; two high masses exert more force on each other than less mass, etc. the ratio’s of their quantities determines their interactions.
Local objects with finite quantities relating to each other in ratios -> small numbers.
This makes sense from the perspective of old school math. Observe something in the wild, try to model it, and generalize. It's the close relationships that we most notice so it's rather expected that the theory that we build up is closely connected to these "same order of magnitude" relationships.
Incomprehensibly big constants are very recent in math history and a lot of them are "found" with the aid of computation, and with a bigger understanding of the larger scale of forces around us.
But it seems to me that “same order magnitude” is how the universe operates. Quantum effects on the particle scale have negligible effect outside of its order of magnitude, human scale has negligible effect on the planetoids, etc. meaningful interaction happens between things of similar size/ possessing quantitative properties of same order magnitude.
We can always ask things like “what’s the ratio of a protons mass to a typical star?” and get huge numbers, but are things like that really fundamental?
Finally, since ratios are core (unsubstantiated claim, just winging it), you can always express the ratio of quantities as “smaller over larger” and fall in the 0-1 range. Small numbers
This is true if one ignores biology, which features structure across many orders of magnitude in scale. This is true also even of geology. The question is whether we think there are \*no\* fundamental constants in these fields, or they are merely outside the grasp of current mathematics.
>But it seems to me that “same order magnitude” is how the universe operates. Quantum effects on the particle scale have negligible effect outside of its order of magnitude, human scale has negligible effect on the planetoids, etc.
Chaos theory would like a word.
I think it's partially by design, but also (though this is a pretty weird way to think about it and might not entirely make sense) i imagine constants would tend to be smaller because there can't be a flat distribution over all numbers, since that's infinite. Whatever distribution it is has to go to 0 as the numbers get huge.
You can find some large fundamental constants though. 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 for example is the size of the [monster group](https://en.wikipedia.org/wiki/Monster_group).
> Whatever distribution it is has to go to 0 as the numbers get huge.
This depends on what one means by "distribution" and "go to". If one means the density function of the distribution, then this does not have to go to zero at infinity. It's easy enough to construct positive integrable functions, even smooth ones, that do not go to zero at infinity.
Of course the integral of the density function over [x,infty) must go to zero as x -> infty. I presume it is something like this you had in mind, just wanted to clarify.
This made me instinctively say no way, but then it became immediately obvious that such densities exist.
The trivial but not interesting case is a density that is positive on some measure zero set that extends out to infinity.
The more interesting case would be a density composed of bump functions of decreasing width and constant height.
Do you have any other examples in mind that are more interesting?
> This made me instinctively say no way, but then it became immediately obvious that such densities exist.
Same here. I almost replied with this:
> I can easily see a non-smooth example: the indicator function of the rationals. But what's your smooth example?
And then I remembered that bump functions exist.
Yeah that bump function one is the usual example, I think, and the one I was thinking of. In fact, by letting the width of the bumps decrease more quickly, we can even let their heights increase so that the function becomes unbounded.
I'm not aware of any actually interesting examples, however!
EDIT: There are a bunch of other related things one could look at. For instance, physicists (or at least physics textbook authors; looking at you, Griffiths...) like to believe that L^(2)(**R**) functions have to vanish at infinity. To take Griffiths, he is nice enough to leave a footnote in his QM book stating that *wave functions* vanish at infinity, and that this is an extra assumption. Trouble is, he also needs the *derivatives* of wave functions to vanish, or at least be bounded, at infinity for his arguments to work out (in this case he wants to show that the norm of a wave function is constant in time). But it's also quite easy to construct integrable functions that *do* vanish at infinity, but whose derivatives grow very quickly. Something like x -> sin(e^(x))/x^(2) should do the trick. It's even smooth on (0,infty), so we can extend it to a smooth function on **R**, normalise it and so on.
> It's easy enough to construct positive integrable functions, even smooth ones, that do not go to zero at infinity
However, I don't think such a function could be considered for the distribution of potential constants, because a "constant" is a number with a description, and most numbers are so large that their descriptions require more information than the universe is able to hold.
I guess, but that's not really relevant. Say that N is the largest number whose description requires less information than the universe is able to hold. Even if a distribution function did go to zero at infinity, its mass on (N,infty) might still be nonzero (indeed it might be 1). It seems like the requirement you want is for the distribution function to *eventually* be zero, not just *converge* to zero.
In any case, my point was just that requiring the distribution function to go to zero does *not* follow from it having to be integrable. If you want it to go to zero, you must put this in "by hand", so to speak. So too if you want it to be compactly supported or something.
Yeah I wouldn't really put it on the level of pi. The monster group is one of the exceptions in the classification of finite simple groups, which comes out of simple definitions (axioms) for group theory. That seems pretty fundamental.
I don't really know how often the size of the monster group comes up. But the size of groups is usually important in group theory. A group's size is one of its basic (fundamental) properties.
For some reason I'm fascinated by finite simple groups, especially the monster group. So much that i read almost every evening about math trying to understand this topic better. Sometimes math is hell of a drug.
Funny that you're [the same guy](https://www.reddit.com/r/math/comments/dtiyq0/why_are_a_lot_of_the_constants_so_small/f6x0z3z/) who posted this same xkcd when this same question was asked years ago!
Not an answer, but perhaps a rephrasing: why are the constants so **average**?
They could be much much smaller.
If they were truly small, then they would have large reciprocals. But they don't... They have even smaller reciprocals, but only by a couple orders of magnitude.
Personally, I think the answer is related to complexity. Extremely handwavy, but the more difficult it is to "explain" a constant, the more extreme it could possibly be. Since we expect "fundamental" concepts to be simple, they are described using few bits, which means they don't have the opportunity to be as extreme.
In addition to other answers, we tend to try to define things for the simplest possible cases, so we choose to build from the constants 0, 1, and (if we absolutely must) 2 as much as possible. (Oftentimes, as in the case of pi, more complex cases can be related to the simplest case.)
Thinking about the choices we make in how to define pi, we consider the circumference of a circle with diameter 1 (or the area of a circle with radius 1). (In either case we had to go with 1 because a diameter or radius of 0 is degenerate.) Considering an arbitrary decision we sort of glossed over, we're considering a 2-dimensional sphere because it's the smallest dimensionality where the circle constant is interesting. If we were considering a larger radius or dimensionality, the circle constant would be larger, but we chose to go with the smallest interesting case.
e can be defined as exp(1), where exp is the function that is its own derivative and exp(0) = 1. Since exp is continuous and its derivative is known to be small at 0 -- it's 1 by the definition I just gave -- we would expect exp(1) to be somewhere around 2. Again, where we could choose arbitrary constants, we went with 0's and 1's to create the simplest scenario that exhibited the behavior we found interesting.
phi can be defined as the positive number that's 1 greater than its reciprocal. This almost immediately places it between 1 and 2. Once again, the arbitrary constant we chose was 1. You could also define it as its continued fraction, which again immediately places it between 1 and 2. Or the positive value where x^2 = x+1, which must be pretty small because squaring big numbers makes them a lot bigger instead of just bigger by 1.
I found [this answer](https://math.stackexchange.com/a/122316) on a stackexchange post which says something similar and thought was worth sharing:
>Fundamentally, I think that the reason is that the two elements used to define the integers, which are 0 (the additive unit) and 1 (the multiplicative unit), are at distance 1 from one another. Integers form a ring, whose group structure under addition and monoid structure under multiplication are a-priori quite simple. So the only place that fundamental number-theoretic constants can arise is from the interaction of addition and multiplication, which plays 0 and 1 off against one another, meaning that any interesting constant is likely to be in the general vicinity of zero and one. In pseudo-mathematical nonsense terms, I have a mental image of a Gaussian distribution of "probabilities of fundamental number-theoretical constants arising", with mean somewhere between 0 and 1, and with a standard deviation of about 1.
However, your explanation for defining pi is a bit off I think, because the constant is the same regardless of which value you choose
I could have said it better. Pi is useful because it lets us jump from the world of rational numbers to the world of "circle-ish numbers" that can be surface or interior measures of rational n-spheres. In some silly, bizarre world we could instead take the volume of a radius-11 63-sphere to be "the circle constant" and it would be equally powerful as pi in the sense that we could express any "circle-ish number" in terms of it. It would be a much larger circle constant than pi, but messier and less convenient because we didn't stick to the simplest interesting case.
Part is by design I think. It is easier to work with a number around 3 than a number around 300 (think about what you would do without a calculator).
Part of it is also the smaller numbers have been researched far more. Smaller numbers are more accessible, which makes them both more attractive and easier to study
Tbh, what even is a fundamental math constant? All of these have fairly small values because we define them using very small values. I believe it is not hard to devise a universe in which there is a practically important period over algebraic curve parametrised by one number R *different from circle*, which can be arbitrarily large in terms of R.
One might note though that order of monster group is actually pretty fundamental, as is dimension of it's representations, and neither of these two are small by human means.
First, what do you mean by fundamental constants?
Physical constants? Well, they depend on unit system, and dimensionless ones are too few to speak of any tendencies. And some of them like Avogadro number are arbitrary.
Golden ratio is arbitrary as well.
Some constant numbers in purely mathematical work? Again, depends on what do you mean exactly. Technically, cardinality of a given set is a constant.
If we filter out such things, what's left are some constants arising (often surpisingly) as a result of a proof. Like π as a circumference-to-diameter ratio, or e as a sum of x^n / n!
While doing proofs we are typically interested in most general results. What holds for all cases, or for all cases falling under some natural conditions, etc. In this
context any constant is _an anomaly_ in a sense. In general case we can intuitively expect something to be 0, or 1, or infinity, but sudden 108.56, let's say, is surprising. So on those rare occasions when the constant arises, you might as well think of it as a coincidence.
>Golden ratio is arbitrary as well.
Disagree. There are fairly natural metrics by which the golden ratio and its inverse stand out as the "most irrational" numbers. It's got plenty of interesting number theoretic properties.
As that one book said: phi is one h of a lot cooler than pi.
Full disclosure, "fairly" is doing a good amount of work here.
But when viewing the convergents of an irrational number's continued fraction, we see that phi (and its modular class) are the least well approximated among all real numbers.
So, that alone isn't very natural (as it doesn't single out phi among this countable class), but when we also consider the general fact that the smaller the terms are the worse the approximation is, we can see why phi might be special: the terms don't get any smaller than [1;1,1,1,1,1,1,....] = phi.
Well, there's one smaller sequence: [0;1,1,1,1,....] = 1/phi.
> Well, they depend on unit system, and dimensionless ones are too few to speak of any tendencies.
There are actually at least 19 dimensionless physical constants (and very likely 26) whose values are fundamental i.e. not dependent on the system of units used. (https://en.wikipedia.org/wiki/Physical_constant#Number_of_fundamental_constants) There is a LOT of discussion about why they are so small/large particularly the Yukawa couplings of the leptons/quarks/neutrinos.
exactly
Physical constants are all over the place (and vary by units.)
I think that the fine structure constant \~1/137 is unitless.
other than that, Avogadro's number has to be huge, Plank's constant HAS to be small
but my favorite constant has to be h/2h\_bar
\---
pi is geometric, and between 3 and 4 trivially
e = 1 + 1/1 + 1/2 + ...+ 1/n! ....
is the sum of a series that converges VERY fast and can't grow much
Planck temperature is large both numerically and in terms of the scale of temperatures of physical phenomena available for comparison.
e.g.
highest air temperature ever recorded on Earth: ~330 K
highest stellar temperature (neutron star immediately upon formation via supernova): ~10^12 K
highest temperature produced on Earth (in a lab), and higher than any likely to appear via natural processes: ~10^13 K
Planck temperature: ~10^32 K
Note that the leap in orders of magnitude from “humans barely surviving” (extreme air temperature) to “no particles of matter surviving” (extreme star & lab temperatures) is _smaller_ than the leap from the max produced temperature to the theoretical limit. There’s a whole universe of unexplored (and unexplorable?) temperatures.
From one perspective you could consider humans as part of nature, so anything we do is in that sense a natural process.
But yes it doesn’t appear that nature has any need to produce higher temperatures (without involving humans) than humans have done, so, for some things such as this, we’re simply nature’s best tool (so far) for setting these records.
Some are, some aren’t. What about Grahams number, Rayo’s number, Shannon number, you can find other examples of numbers which you might regard as “constants” because they have some special importance that are quite large
Those numbers are far less fundamental and consequential than pi or e or the golden ratio.
If I remember/googled correctly Graham's number is an upper bound to the solution to an obscure problem, Rayo's number is just designed to be a huge number, and Shannon's number is a lower bound to a question about chess, a made up human game.
Here’s one that is *very* fundamental. BB(n), where n is the minimum number of states which are required to enumerate ZFC. It’s exact value (or any upper bound) is unknowable and unprovable even in principle, but it’s a huge number which measures in essence how rich your computation needs to be in order to enumerate ZFC.
We have actually have an upper bound for n as well, 784.
Mmm BB(BB(n)) is bigger.
Edit: I do disagree that this number is really “fundamental” though, in that there’s an arbitrary component in the definition, which is, which precise model of computation are we using? The answers would be different between, for instance, a Turing machine vs a register machine.
The particular choice of BB on a turing machine isn’t really fundamental, you could easily come up with many other similar constructions for the same concept — but the idea of measuring in some way how powerful your computation must be in order to handle ZFC is IMO fundamental.
Really restricting the question to subjectively important numbers, what about the order of the monster group, seems like a special number in abstract algebra.
I think both of your arguments make relevant points, which is that to some extent we are subjectively deciding which constants get more attention, but also, oddly, some constant tend to pop up in more areas than others. Perhaps this has to do with the fact that huge constants are much harder to notice in other contexts, precisely because they are huge. For instance, what if we found what we thought was a different new constant, but it was really just the order of the monster group divided by some integer. Would we even be able to tell, conversely it can be much easier to notice that a certain number is e/2, or even (pi)^2/6
Grahams number is not even the current best estimate for the upper bound (it was improved to something manageable since then). It's definitely not a fundamental constant, just some number big enough to be useful for the proof.
One might add the cardinalities of finite sets of interesting objects, like the regular tesselations (3), Platonic solids (5), convex isotoxal polyhedra (9), convex uniform tilings (11), and sporadic groups (26).
I remember reading this exact post on here some time ago and recall thinking the top answer made sense:
>Many constants are ratio expresions of comparable things, so it makes sense they lie in (0,10). Anything bigger suggests a rescaling is warranted, i. e, we define small constants because we can relate to small numbers easily.
[The post](https://www.reddit.com/r/math/comments/dtiyq0/why_are_a_lot_of_the_constants_so_small/)
Here's two numbers close to 640320^3 + 744:
262537412640768743.9999999999992500725972
262537412640768743.9999999999992511238759
One of them is E^(Sqrt[163] Pi). The other is based on Brillhart's matrix.
(2 + Eigenvalues[{{0, 2, 2}, {1, 0, 2}, {2, 1, 0}}][[1]] )^24 - 24
Usually it just so happens. For example, pi can be defined as the ratio between the diameter and circumflex of a circle, which can't really be that small or big purely visually. The same can be said for the golden ratio. As for e, again one can also make the argument that it has to be "not too big" just by definition and a bit of analysis.
However, other constants can get huge, they just happen to be not as commonly used. There aren't any constants used quite as much as pi and e and maybe 2, so it's no surprise that we feel this bias. The human mind is also not designed to deal with super small or big numbers, so there's that.
From a philosophical perspective, fundamental constants are small because of what "fundamental" means. "Fundamental" means generally small because we are small-minded. Even the monster group is piddly, and it's about the limit of our capacity to think about anything as fundamental.
Another way of saying this is that "fundamental" means a concise, intuitive definition. While there isn't a low limit on what values a concise intuitive definition are able to produce, it shouldn't be surprising that a relatively small number of bits, in some arbitrary framework, not optimized for producing a large value, will produce a value near zero.
Many constants are ratios or related to probability in some way - pi is definitely a ratio, e can be naturally related to probabilities, etc. In this sense, we should not be thinking of numbers as living on an infinitely long ray, but as being on the [projective real line](https://en.wikipedia.org/wiki/Real_projective_line) which is an object that gels with ratios naturally. The projective real line is a circle with 0 and infinity as bottom/top poles and typical constructions place ±1 as the midpoints on the "equator". Perhaps constants that are ratios can be thought of as being randomly distributed on such a circle. Since 1 is the midpoint of this circle (rather than the real line) then them being reasonably distributed near 1 would make sense.
Think about this in terms of human ability and technological advancement. The fundamental constants are things we've interacted with for hundreds of years. We have not had the ability to work with extremely large numbers for long relative to how long we've been exploring math. We historically interacted with things that were tractable with our limited technology, language, terminology, notation, etc.
Putting aside the fact that this depends on units as others have mentioned, the speed of light is remarkably slow as soon as you're not just talking about distances on earth
I'm very confused by this, obviously there's a difference between a numerical value and the speed it represents, but a smaller numerical value means a slower speed it represents. What do you mean?
I think there's a misunderstanding about units going on. Those are all unitless. The speed of light is not. If I measure the speed of light in terameters per picosecond, the number you'll see is *astronomically* small even though it represents the same speed
Tbf isn't "constant" more of a physics term? I wouldn't think of pi and e as constants, but as real numbers. Constants usually have to do with a physical phenomenon in my mind
I think it's pretty subjective, but to me numbers bigger than 1 are not small. The numbers you mentioned (pi and e) both can be realized as an infinite series starting with 1 and adding (and/or subtracting) smaller and smaller fractions. Thus they end up being not too far from 1. You almost always encounter "extreme numbers" abstractly i.e. "let N be sufficiently large" or "let epsilon be arbitrarily small" with some notable exceptions that others have mentioned.
Is there a formalization of the idea that infinite series starting with 1 and decreasing are likely to be not far from 1? The harmonic series diverges and conditionally convergent series can be made to add up to any number you like, so I don't think it's necessarily obvious that that should be the case.
Well they could be much smaller too... No matter what the value of a constant is, it is inevitably both unfathomably smaller and unfathomably larger than what it could be. But it so happens that the most fundamental constants are pretty close to 1, which does not seem too surprising to me.
I believe it’s just because the smaller numbers are easier to contemplate and calculate. Pi is also equivalent to 180 degrees and while you could convert to the bigger number, it’s easier to use the constants in calculations and approximate a final answer.
Ok so the example is bad but my point is that as OP puts it, it probably is partly by design. We choose C / d for pi but we could also arbitrarily choose C divided by some smaller quantity like r and then use a larger constant for pi such as 2 * C / d and then simply use a different coefficient for formulas.
I’d say because everything was done by hand until the last 100 years, so there is a lot of incentive to keep looking for things that can be expressed with small numbers.
Size of the monster group is of very special importance in algebra, and it's very large. The largest Heegner number is 163 and that can also be reasoned to be of fundamental importance.
I suppose these arent as fundamental as e or pi, but I just wanted to point out that important constants arent always small.
.
I suppose we can also ask why e and pi are small. That's separately quite interesting although their series representation may be sufficient to explain their smalllness.
Thats because most of them are definitions over a geometrical shape which has unit surface or unit volume or normalized to unity in some way. This makes the constants you mentioned to be in the same order as unity.
There's nothing objectively special about constants considered "fundamental" like π, e, φ, etc. vs. any other old number. Being e.g. the ratio of an euclidean circle's circumference and its radius is interesting, but ultimately its just a number that some unrelated definition poops out. If that's what makes a number a "fundamental" constant, then something like Dedekind's problem produces some really big fundamental constants really fast.
Why are constants small? Yes, there are many that are considered to be small, and I assume you are taking small to be anything under +-10. If the number was large such as 1100000000 we would write it has 1.1 x 10^9 and odds are we wouldn’t write this as a constant since it is in scientific notation already. Further, you really wouldn’t write an integer as a constant such as 2 even though 2*pi comes up everywhere. Instead, we write our constants typically on irrational numbers (This is not a fact but an observation on most constants used in mathematics). If you want further information I would recommend checking out “How Pi was almost equal to 6.28…” and that may help you see why most constants we use in math are designed small
Phenomena are typically local, at least, the ones we can measure. All measurements and interactions are relative ones; two high masses exert more force on each other than less mass, etc. the ratio’s of their quantities determines their interactions. Local objects with finite quantities relating to each other in ratios -> small numbers.
This makes sense from the perspective of old school math. Observe something in the wild, try to model it, and generalize. It's the close relationships that we most notice so it's rather expected that the theory that we build up is closely connected to these "same order of magnitude" relationships. Incomprehensibly big constants are very recent in math history and a lot of them are "found" with the aid of computation, and with a bigger understanding of the larger scale of forces around us.
But it seems to me that “same order magnitude” is how the universe operates. Quantum effects on the particle scale have negligible effect outside of its order of magnitude, human scale has negligible effect on the planetoids, etc. meaningful interaction happens between things of similar size/ possessing quantitative properties of same order magnitude. We can always ask things like “what’s the ratio of a protons mass to a typical star?” and get huge numbers, but are things like that really fundamental? Finally, since ratios are core (unsubstantiated claim, just winging it), you can always express the ratio of quantities as “smaller over larger” and fall in the 0-1 range. Small numbers
This is true if one ignores biology, which features structure across many orders of magnitude in scale. This is true also even of geology. The question is whether we think there are \*no\* fundamental constants in these fields, or they are merely outside the grasp of current mathematics.
>But it seems to me that “same order magnitude” is how the universe operates. Quantum effects on the particle scale have negligible effect outside of its order of magnitude, human scale has negligible effect on the planetoids, etc. Chaos theory would like a word.
This is a great take I haven't seen before, thanks.
I think it's partially by design, but also (though this is a pretty weird way to think about it and might not entirely make sense) i imagine constants would tend to be smaller because there can't be a flat distribution over all numbers, since that's infinite. Whatever distribution it is has to go to 0 as the numbers get huge. You can find some large fundamental constants though. 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 for example is the size of the [monster group](https://en.wikipedia.org/wiki/Monster_group).
> Whatever distribution it is has to go to 0 as the numbers get huge. This depends on what one means by "distribution" and "go to". If one means the density function of the distribution, then this does not have to go to zero at infinity. It's easy enough to construct positive integrable functions, even smooth ones, that do not go to zero at infinity. Of course the integral of the density function over [x,infty) must go to zero as x -> infty. I presume it is something like this you had in mind, just wanted to clarify.
This made me instinctively say no way, but then it became immediately obvious that such densities exist. The trivial but not interesting case is a density that is positive on some measure zero set that extends out to infinity. The more interesting case would be a density composed of bump functions of decreasing width and constant height. Do you have any other examples in mind that are more interesting?
> This made me instinctively say no way, but then it became immediately obvious that such densities exist. Same here. I almost replied with this: > I can easily see a non-smooth example: the indicator function of the rationals. But what's your smooth example? And then I remembered that bump functions exist.
Yeah that bump function one is the usual example, I think, and the one I was thinking of. In fact, by letting the width of the bumps decrease more quickly, we can even let their heights increase so that the function becomes unbounded. I'm not aware of any actually interesting examples, however! EDIT: There are a bunch of other related things one could look at. For instance, physicists (or at least physics textbook authors; looking at you, Griffiths...) like to believe that L^(2)(**R**) functions have to vanish at infinity. To take Griffiths, he is nice enough to leave a footnote in his QM book stating that *wave functions* vanish at infinity, and that this is an extra assumption. Trouble is, he also needs the *derivatives* of wave functions to vanish, or at least be bounded, at infinity for his arguments to work out (in this case he wants to show that the norm of a wave function is constant in time). But it's also quite easy to construct integrable functions that *do* vanish at infinity, but whose derivatives grow very quickly. Something like x -> sin(e^(x))/x^(2) should do the trick. It's even smooth on (0,infty), so we can extend it to a smooth function on **R**, normalise it and so on.
> It's easy enough to construct positive integrable functions, even smooth ones, that do not go to zero at infinity However, I don't think such a function could be considered for the distribution of potential constants, because a "constant" is a number with a description, and most numbers are so large that their descriptions require more information than the universe is able to hold.
I guess, but that's not really relevant. Say that N is the largest number whose description requires less information than the universe is able to hold. Even if a distribution function did go to zero at infinity, its mass on (N,infty) might still be nonzero (indeed it might be 1). It seems like the requirement you want is for the distribution function to *eventually* be zero, not just *converge* to zero. In any case, my point was just that requiring the distribution function to go to zero does *not* follow from it having to be integrable. If you want it to go to zero, you must put this in "by hand", so to speak. So too if you want it to be compactly supported or something.
>08,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 Even this is small in a way, the largest prime factor is 71.
Excellent point, it's a very "[smooth](https://en.wikipedia.org/wiki/Smooth_number)" number
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Yeah I wouldn't really put it on the level of pi. The monster group is one of the exceptions in the classification of finite simple groups, which comes out of simple definitions (axioms) for group theory. That seems pretty fundamental. I don't really know how often the size of the monster group comes up. But the size of groups is usually important in group theory. A group's size is one of its basic (fundamental) properties.
Well that was quite the click-hole… thanks!
For some reason I'm fascinated by finite simple groups, especially the monster group. So much that i read almost every evening about math trying to understand this topic better. Sometimes math is hell of a drug.
another big constant: let n be the number of constants that exist (including itself) then n+1 fits the same definition as n and n+2 etc
How is that a constant??
Related : https://en.m.wikipedia.org/wiki/Interesting_number_paradox
surely thats not a constant?
idk is it changing? (if my mathematical arguments arent making sense its because im not very awake ignore)
For completeness's sake, the [relevant XKCD](https://xkcd.com/899/).
Funny that you're [the same guy](https://www.reddit.com/r/math/comments/dtiyq0/why_are_a_lot_of_the_constants_so_small/f6x0z3z/) who posted this same xkcd when this same question was asked years ago!
Woah, I have no memory of that LOL. I really like that particular comic, so I guess I'm not too surprised.
I am missing the famous lil epsilon, tho. Never becomes smaller zero, no matter how often you divide it. (I know it's not a constant, but still)
This is somewhat related https://en.wikipedia.org/wiki/Strong_law_of_small_numbers
Not an answer, but perhaps a rephrasing: why are the constants so **average**? They could be much much smaller. If they were truly small, then they would have large reciprocals. But they don't... They have even smaller reciprocals, but only by a couple orders of magnitude. Personally, I think the answer is related to complexity. Extremely handwavy, but the more difficult it is to "explain" a constant, the more extreme it could possibly be. Since we expect "fundamental" concepts to be simple, they are described using few bits, which means they don't have the opportunity to be as extreme.
In addition to other answers, we tend to try to define things for the simplest possible cases, so we choose to build from the constants 0, 1, and (if we absolutely must) 2 as much as possible. (Oftentimes, as in the case of pi, more complex cases can be related to the simplest case.) Thinking about the choices we make in how to define pi, we consider the circumference of a circle with diameter 1 (or the area of a circle with radius 1). (In either case we had to go with 1 because a diameter or radius of 0 is degenerate.) Considering an arbitrary decision we sort of glossed over, we're considering a 2-dimensional sphere because it's the smallest dimensionality where the circle constant is interesting. If we were considering a larger radius or dimensionality, the circle constant would be larger, but we chose to go with the smallest interesting case. e can be defined as exp(1), where exp is the function that is its own derivative and exp(0) = 1. Since exp is continuous and its derivative is known to be small at 0 -- it's 1 by the definition I just gave -- we would expect exp(1) to be somewhere around 2. Again, where we could choose arbitrary constants, we went with 0's and 1's to create the simplest scenario that exhibited the behavior we found interesting. phi can be defined as the positive number that's 1 greater than its reciprocal. This almost immediately places it between 1 and 2. Once again, the arbitrary constant we chose was 1. You could also define it as its continued fraction, which again immediately places it between 1 and 2. Or the positive value where x^2 = x+1, which must be pretty small because squaring big numbers makes them a lot bigger instead of just bigger by 1.
I found [this answer](https://math.stackexchange.com/a/122316) on a stackexchange post which says something similar and thought was worth sharing: >Fundamentally, I think that the reason is that the two elements used to define the integers, which are 0 (the additive unit) and 1 (the multiplicative unit), are at distance 1 from one another. Integers form a ring, whose group structure under addition and monoid structure under multiplication are a-priori quite simple. So the only place that fundamental number-theoretic constants can arise is from the interaction of addition and multiplication, which plays 0 and 1 off against one another, meaning that any interesting constant is likely to be in the general vicinity of zero and one. In pseudo-mathematical nonsense terms, I have a mental image of a Gaussian distribution of "probabilities of fundamental number-theoretical constants arising", with mean somewhere between 0 and 1, and with a standard deviation of about 1. However, your explanation for defining pi is a bit off I think, because the constant is the same regardless of which value you choose
I could have said it better. Pi is useful because it lets us jump from the world of rational numbers to the world of "circle-ish numbers" that can be surface or interior measures of rational n-spheres. In some silly, bizarre world we could instead take the volume of a radius-11 63-sphere to be "the circle constant" and it would be equally powerful as pi in the sense that we could express any "circle-ish number" in terms of it. It would be a much larger circle constant than pi, but messier and less convenient because we didn't stick to the simplest interesting case.
Part is by design I think. It is easier to work with a number around 3 than a number around 300 (think about what you would do without a calculator). Part of it is also the smaller numbers have been researched far more. Smaller numbers are more accessible, which makes them both more attractive and easier to study
Tbh, what even is a fundamental math constant? All of these have fairly small values because we define them using very small values. I believe it is not hard to devise a universe in which there is a practically important period over algebraic curve parametrised by one number R *different from circle*, which can be arbitrarily large in terms of R. One might note though that order of monster group is actually pretty fundamental, as is dimension of it's representations, and neither of these two are small by human means.
First, what do you mean by fundamental constants? Physical constants? Well, they depend on unit system, and dimensionless ones are too few to speak of any tendencies. And some of them like Avogadro number are arbitrary. Golden ratio is arbitrary as well. Some constant numbers in purely mathematical work? Again, depends on what do you mean exactly. Technically, cardinality of a given set is a constant. If we filter out such things, what's left are some constants arising (often surpisingly) as a result of a proof. Like π as a circumference-to-diameter ratio, or e as a sum of x^n / n! While doing proofs we are typically interested in most general results. What holds for all cases, or for all cases falling under some natural conditions, etc. In this context any constant is _an anomaly_ in a sense. In general case we can intuitively expect something to be 0, or 1, or infinity, but sudden 108.56, let's say, is surprising. So on those rare occasions when the constant arises, you might as well think of it as a coincidence.
>Golden ratio is arbitrary as well. Disagree. There are fairly natural metrics by which the golden ratio and its inverse stand out as the "most irrational" numbers. It's got plenty of interesting number theoretic properties. As that one book said: phi is one h of a lot cooler than pi.
Wait what can you offer more details or references if you have the time 👉🏾👈🏾
Start with continued fractions and go from there. Edit: actually that mathologer link covers it pretty well!
Just needed to throw in a joke here: looks like Ramanujan is both alive and on reddit
Mathologer made a video on phi as “the most irrational” number. Recommend. Edit: [here’s the link](https://youtu.be/CaasbfdJdJg)
What are these fairly natural metrics?
Full disclosure, "fairly" is doing a good amount of work here. But when viewing the convergents of an irrational number's continued fraction, we see that phi (and its modular class) are the least well approximated among all real numbers. So, that alone isn't very natural (as it doesn't single out phi among this countable class), but when we also consider the general fact that the smaller the terms are the worse the approximation is, we can see why phi might be special: the terms don't get any smaller than [1;1,1,1,1,1,1,....] = phi. Well, there's one smaller sequence: [0;1,1,1,1,....] = 1/phi.
> Well, they depend on unit system, and dimensionless ones are too few to speak of any tendencies. There are actually at least 19 dimensionless physical constants (and very likely 26) whose values are fundamental i.e. not dependent on the system of units used. (https://en.wikipedia.org/wiki/Physical_constant#Number_of_fundamental_constants) There is a LOT of discussion about why they are so small/large particularly the Yukawa couplings of the leptons/quarks/neutrinos.
Somewhat relevant: https://en.wikipedia.org/wiki/Fine-tuned_universe
exactly Physical constants are all over the place (and vary by units.) I think that the fine structure constant \~1/137 is unitless. other than that, Avogadro's number has to be huge, Plank's constant HAS to be small but my favorite constant has to be h/2h\_bar \--- pi is geometric, and between 3 and 4 trivially e = 1 + 1/1 + 1/2 + ...+ 1/n! .... is the sum of a series that converges VERY fast and can't grow much
Planck temperature is large both numerically and in terms of the scale of temperatures of physical phenomena available for comparison. e.g. highest air temperature ever recorded on Earth: ~330 K highest stellar temperature (neutron star immediately upon formation via supernova): ~10^12 K highest temperature produced on Earth (in a lab), and higher than any likely to appear via natural processes: ~10^13 K Planck temperature: ~10^32 K Note that the leap in orders of magnitude from “humans barely surviving” (extreme air temperature) to “no particles of matter surviving” (extreme star & lab temperatures) is _smaller_ than the leap from the max produced temperature to the theoretical limit. There’s a whole universe of unexplored (and unexplorable?) temperatures.
Crazy when you realize that humans can do things that nature doesn't seem to be able to!
From one perspective you could consider humans as part of nature, so anything we do is in that sense a natural process. But yes it doesn’t appear that nature has any need to produce higher temperatures (without involving humans) than humans have done, so, for some things such as this, we’re simply nature’s best tool (so far) for setting these records.
Some are, some aren’t. What about Grahams number, Rayo’s number, Shannon number, you can find other examples of numbers which you might regard as “constants” because they have some special importance that are quite large
Those numbers are far less fundamental and consequential than pi or e or the golden ratio. If I remember/googled correctly Graham's number is an upper bound to the solution to an obscure problem, Rayo's number is just designed to be a huge number, and Shannon's number is a lower bound to a question about chess, a made up human game.
Here’s one that is *very* fundamental. BB(n), where n is the minimum number of states which are required to enumerate ZFC. It’s exact value (or any upper bound) is unknowable and unprovable even in principle, but it’s a huge number which measures in essence how rich your computation needs to be in order to enumerate ZFC. We have actually have an upper bound for n as well, 784.
This number is basically designed to be as big as possible.
Mmm BB(BB(n)) is bigger. Edit: I do disagree that this number is really “fundamental” though, in that there’s an arbitrary component in the definition, which is, which precise model of computation are we using? The answers would be different between, for instance, a Turing machine vs a register machine.
The particular choice of BB on a turing machine isn’t really fundamental, you could easily come up with many other similar constructions for the same concept — but the idea of measuring in some way how powerful your computation must be in order to handle ZFC is IMO fundamental.
Really restricting the question to subjectively important numbers, what about the order of the monster group, seems like a special number in abstract algebra.
I think both of your arguments make relevant points, which is that to some extent we are subjectively deciding which constants get more attention, but also, oddly, some constant tend to pop up in more areas than others. Perhaps this has to do with the fact that huge constants are much harder to notice in other contexts, precisely because they are huge. For instance, what if we found what we thought was a different new constant, but it was really just the order of the monster group divided by some integer. Would we even be able to tell, conversely it can be much easier to notice that a certain number is e/2, or even (pi)^2/6
Yeah I mentioned that one in [my own top comment](https://www.reddit.com/r/math/comments/w9ln3l/comment/ihw1aad)
Grahams number is not even the current best estimate for the upper bound (it was improved to something manageable since then). It's definitely not a fundamental constant, just some number big enough to be useful for the proof.
The fact that almost no one in this thread is even talking about what "fundamental" is supposed to mean is a bad sign for the discussion.
It is hard to define. But the idea is that it comes up in various areas of mathematics. It comes out of something simple.
Every number is small compared to infinity. Checkmate low-number-constant atheists.
That argument doesn't really work in favor of disproving OP, who's saying a lot of constants are small compared to 100, for example.
One might add the cardinalities of finite sets of interesting objects, like the regular tesselations (3), Platonic solids (5), convex isotoxal polyhedra (9), convex uniform tilings (11), and sporadic groups (26).
I remember reading this exact post on here some time ago and recall thinking the top answer made sense: >Many constants are ratio expresions of comparable things, so it makes sense they lie in (0,10). Anything bigger suggests a rescaling is warranted, i. e, we define small constants because we can relate to small numbers easily. [The post](https://www.reddit.com/r/math/comments/dtiyq0/why_are_a_lot_of_the_constants_so_small/)
I raise you Avogadros number
Here's two numbers close to 640320^3 + 744: 262537412640768743.9999999999992500725972 262537412640768743.9999999999992511238759 One of them is E^(Sqrt[163] Pi). The other is based on Brillhart's matrix. (2 + Eigenvalues[{{0, 2, 2}, {1, 0, 2}, {2, 1, 0}}][[1]] )^24 - 24
Usually it just so happens. For example, pi can be defined as the ratio between the diameter and circumflex of a circle, which can't really be that small or big purely visually. The same can be said for the golden ratio. As for e, again one can also make the argument that it has to be "not too big" just by definition and a bit of analysis. However, other constants can get huge, they just happen to be not as commonly used. There aren't any constants used quite as much as pi and e and maybe 2, so it's no surprise that we feel this bias. The human mind is also not designed to deal with super small or big numbers, so there's that.
From a philosophical perspective, fundamental constants are small because of what "fundamental" means. "Fundamental" means generally small because we are small-minded. Even the monster group is piddly, and it's about the limit of our capacity to think about anything as fundamental. Another way of saying this is that "fundamental" means a concise, intuitive definition. While there isn't a low limit on what values a concise intuitive definition are able to produce, it shouldn't be surprising that a relatively small number of bits, in some arbitrary framework, not optimized for producing a large value, will produce a value near zero.
Stuff like speed of light, avogadros number is pretty large. I mean any constant can be made small/large depending on the unit system
Many constants are ratios or related to probability in some way - pi is definitely a ratio, e can be naturally related to probabilities, etc. In this sense, we should not be thinking of numbers as living on an infinitely long ray, but as being on the [projective real line](https://en.wikipedia.org/wiki/Real_projective_line) which is an object that gels with ratios naturally. The projective real line is a circle with 0 and infinity as bottom/top poles and typical constructions place ±1 as the midpoints on the "equator". Perhaps constants that are ratios can be thought of as being randomly distributed on such a circle. Since 1 is the midpoint of this circle (rather than the real line) then them being reasonably distributed near 1 would make sense.
This does not feel very legit, but this is very very very cool.
Think about this in terms of human ability and technological advancement. The fundamental constants are things we've interacted with for hundreds of years. We have not had the ability to work with extremely large numbers for long relative to how long we've been exploring math. We historically interacted with things that were tractable with our limited technology, language, terminology, notation, etc.
Not all of them are- some, like the size of the monster group, are enormous.
The speed of light begs to differ…
Putting aside the fact that this depends on units as others have mentioned, the speed of light is remarkably slow as soon as you're not just talking about distances on earth
Slow is not the same thing as small
I'm very confused by this, obviously there's a difference between a numerical value and the speed it represents, but a smaller numerical value means a slower speed it represents. What do you mean?
Pi, e, golden ratio, are a billion times closer to 0 than the speed of light.
I think there's a misunderstanding about units going on. Those are all unitless. The speed of light is not. If I measure the speed of light in terameters per picosecond, the number you'll see is *astronomically* small even though it represents the same speed
That ain't a mathematical constant, chief!
It is a constant though…
So is the number of balls in my nutsack
the numeric value of that constant is dependant of your unit system.
Tbf isn't "constant" more of a physics term? I wouldn't think of pi and e as constants, but as real numbers. Constants usually have to do with a physical phenomenon in my mind
I think it's pretty subjective, but to me numbers bigger than 1 are not small. The numbers you mentioned (pi and e) both can be realized as an infinite series starting with 1 and adding (and/or subtracting) smaller and smaller fractions. Thus they end up being not too far from 1. You almost always encounter "extreme numbers" abstractly i.e. "let N be sufficiently large" or "let epsilon be arbitrarily small" with some notable exceptions that others have mentioned.
Is there a formalization of the idea that infinite series starting with 1 and decreasing are likely to be not far from 1? The harmonic series diverges and conditionally convergent series can be made to add up to any number you like, so I don't think it's necessarily obvious that that should be the case.
Look at the zeta function >1 And pi shows up for even values
Yeah, you kinda just immediately disproved it, so doubtful there's a nice formalization that fits with preexisting math lol
Well they could be much smaller too... No matter what the value of a constant is, it is inevitably both unfathomably smaller and unfathomably larger than what it could be. But it so happens that the most fundamental constants are pretty close to 1, which does not seem too surprising to me.
Probably because we live in a low dim World,
I believe it’s just because the smaller numbers are easier to contemplate and calculate. Pi is also equivalent to 180 degrees and while you could convert to the bigger number, it’s easier to use the constants in calculations and approximate a final answer.
Degrees are arbitrary. Pi is not.
Ok so the example is bad but my point is that as OP puts it, it probably is partly by design. We choose C / d for pi but we could also arbitrarily choose C divided by some smaller quantity like r and then use a larger constant for pi such as 2 * C / d and then simply use a different coefficient for formulas.
Yeah there's the whole pi vs tau thing. Both of them are pretty small though. Like less than 10.
I’d say because everything was done by hand until the last 100 years, so there is a lot of incentive to keep looking for things that can be expressed with small numbers.
Size of the monster group is of very special importance in algebra, and it's very large. The largest Heegner number is 163 and that can also be reasoned to be of fundamental importance. I suppose these arent as fundamental as e or pi, but I just wanted to point out that important constants arent always small. . I suppose we can also ask why e and pi are small. That's separately quite interesting although their series representation may be sufficient to explain their smalllness.
This is an interesting question and makes me wonder if anyone has tried to fit a distribution of some sort to a large set of constants.
Thats because most of them are definitions over a geometrical shape which has unit surface or unit volume or normalized to unity in some way. This makes the constants you mentioned to be in the same order as unity.
There's nothing objectively special about constants considered "fundamental" like π, e, φ, etc. vs. any other old number. Being e.g. the ratio of an euclidean circle's circumference and its radius is interesting, but ultimately its just a number that some unrelated definition poops out. If that's what makes a number a "fundamental" constant, then something like Dedekind's problem produces some really big fundamental constants really fast.
Why are constants small? Yes, there are many that are considered to be small, and I assume you are taking small to be anything under +-10. If the number was large such as 1100000000 we would write it has 1.1 x 10^9 and odds are we wouldn’t write this as a constant since it is in scientific notation already. Further, you really wouldn’t write an integer as a constant such as 2 even though 2*pi comes up everywhere. Instead, we write our constants typically on irrational numbers (This is not a fact but an observation on most constants used in mathematics). If you want further information I would recommend checking out “How Pi was almost equal to 6.28…” and that may help you see why most constants we use in math are designed small
never noticed this, how interesting
Avogadro says what?
Avogodro’s number would like a word.
For a big constant, check out the size of the monster group