Why are all the interesting constants so small?

A quick look at the wikipedia entry on mathematical constants suggests that the most important fundamental constants all live in the immediate neighborhood of the first few positive integers. Is there some kind of normalization going on, or some other reasonable explanation for why we have only identified interesting small constants?

EDIT: I may have been too strong in some of my language, or unclear in my examples. The most “important” or “interesting” constants are certainly debatable. Moreover, there are many important and interesting very large numbers. Therefor I would like to make two revisions.

First, to give a clearer idea of the numbers I had in mind, please consider such examples as $\pi$, $e$, the golden ratio, the Euler–Mascheroni constant, the Feigenbaum constants, the twin prime constant, etc. Obviously numbers like $0$, $1$, $\sqrt2$, $…$, while on the wikipedia list, are in some sense “too fundamental” for consideration.

This leads me to my second revision, which is that the constants I am trying to describe are (or appear to be) irrational. Perhaps this is a clue to what makes them interesting. At the very least, it leads me to believe that large integer counterexamples do not satisfy the question as I had intended.

Finally, if I could choose a better word to describe such numbers, it might be “auspicious” rather than interesting or important. But I don’t really know if that’s any better or worse.

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What about the reversible 100 digit prime (i.e. it is a prime if written backwards)

$31399719737866347113914486515772694858917594191229$
$38744591877656925789747974914319422889611373939731$

that breaks up into 10 reversible 10 digit primes in order

$$3139971973, 7866347113, 9144865157, 7269485891, 7594191229, 3874459187, 7656925789, 7479749143, 1942288961, 1373939731$$

which can be assembled into this “magic” square

$$3~~~ 1~~~ 3~~~ 9 ~~~9 ~~~7 ~~~1~~~ 9~~~ 7~~~ 3$$
$$7~~~ 8~~~ 6 ~~~6~~~ 3~~~ 4~~~ 7~~~ 1~~~ 1~~~ 3$$
$$9~~~ 1~~~ 4~~~ 4~~~ 8~~~ 6~~~ 5~~~ 1~~~ 5 ~~~7$$
$$7~~~ 2~~~ 6~~~ 9~~~ 4~~~ 8~~~ 5~~~ 8~~~ 9~~~ 1$$
$$7~~~ 5~~~ 9~~~ 4~~~ 1~~~ 9~~~ 1~~~ 2 ~~~2 ~~~9$$
$$3~~~ 8~~~ 7~~~ 4~~~ 4~~~ 5~~~ 9~~~ 1~~~ 8~~~ 7$$
$$7~~~ 6~~~ 5~~~ 6~~~ 9~~~ 2~~~ 5~~~ 7~~~ 8~~~ 9$$
$$7~~~ 4 ~~~7~~~ 9~~~ 7~~~ 4~~~ 9~~~ 1~~~ 4~~~ 3$$
$$1~~~ 9~~~ 4~~~ 2~~~ 2~~~ 8 ~~~8~~~ 9~~~ 6~~~ 1$$
$$1~~~ 3~~~ 7 ~~~3~~~ 9~~~ 3~~~ 9~~~ 7~~~ 3~~~ 1$$

Where each column row and diagonal is a reversible prime.

If that is not interesting to you, I don’t know what is.

Without waxing too metaphysical, I think that in addition to some “fundamental truths of nature” type answers, there are probably some anthropomorphic reasons partially explaining this observation. We spend most of our waking hours dealing with numbers less than, say, a couple of thousand, so it’s not surprising that most of our most amazing observations concern numbers in this range. It seems likely that as mathematics and technology progress, we will find ourselves discovering amazing properties of ever-increasingly large numbers. Indeed, one of the most amazing numbers ever,

$$808017424794512875886459904961710757005754368000000000,$$

had to wait until the 1970s before its significance was even conjecturally understood. (Edit to add that this number is the order of the monster group, the mathematics behind which couldn’t possibly be properly addressed in this answer. But wikipedia is a good start.)

Also edited to add a response to GregL that was becoming too long for comments. I see your point but ultimately still disagree. It’s hard to make this precise (and so much for not waxing metaphysical), but say we lived in a universe where the ratio of the circumference of a circle to its diameter was on the order of a billion, instead of the current universe’s ratio of $\pi$. Then we might not have ever even noticed that this ratio was constant across all circles, so in a sense it’s only because $\pi$ is small that we were led to observe and hence calculate it. (Okay, $\pi$’s not the best example of this, but you see the point). So in response to a general claim of “The important numbers just turn out to be small when we calculate them,” my answer above is roughly the argument that it’s instead the case that small numbers self-select to even be calculated in the first place! I think the monster group order fits perfectly into this narrative — it is a number representing a tremendous amount of fundamental truth, but it was impossible to know its significance before developing the mathematics and noticing the patterns that forced it to reveal itself.

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 $1$.

The exception which proves the rule, I think, is the circle constant $\tau=2\pi$. I’m of the school of thought that $\tau$;, or the complex number $i\tau$, are more fundamental than $\pi$;. The constant $\tau$, the quotient of the circumference by the radius of the unit circle, lives in the vicinity of $6$, which is a bit far (but not too far) from $0$ and $1$. So $\tau$ is an outlier.

Well, a few observations that may or may not be relevant.

Looking at the numbers in the wikipedia article, a few things pop out.

Many of them are direct relatives of $e$ and $\pi$ and $\sqrt{2}$, so if we accept these as being in the range in question due to pure coincidence, then that explains several others, or at least moves the question towards why these constants are so fundamental.

Several constants concern the difference or ratio of e.g. two series approaching infinity. Arguably, such ratios are interesting mostly when the series in question are somewhat close to each other – if they are vastly different, then whatever actual thing is being examined is likely more obvious to observe and perhaps isn’t deemed worthy of a constant.

Yet other constants are related to limits of power series. These power series then in turn tend to have “small” coefficients – perhaps because we find those functions most “natural”. A related argument would be that the functions and properties we examine behave fairly nicely with regards to order of magnitude – we can reasonably graph most of them on $[0,1]\times[0,1]$ or similar intervals. This could in fact turn into the rather similar question: Why is that so? Clearly, “most” functions $\mathbb{R} \to \mathbb{R}$ will have vastly different values in this range, even when we restrict ourselves to definable, continuous, differentiable or somesuch functions.

And yet another aspect is that often we actually choose the examined property to be in that range because our goal is to show something is “close”. For instance, look at the Euler–Mascheroni constant – it is defined as the (limit of the) difference between two “close” properties.

Lastly, it might also be that such properties are harder to find – for instance, if the limit of a ratio between two series sufficiently differently defined to be not trivial to relate exists, but is a gigantic number, anyone examining and comparing the first few terms might not get the idea that there is a relation of this sort.

In physics the situation is the opposite. To date no theory can provide means to calculate exactly the dimensionless constants arising in Nature, but to explain why they are in the range they are can be done by applying the anthropic principle.

$10^{42}\approx$ electromagnetic / gravitational force of an electron

Your question seems to arise from bias towards addition. There is really no “half way point” between 0 and $\infty$, so you can’t say $\pi$ is “small” in that sense – small compared to what? There’s an uncountably infinite number of numbers smaller than $\pi$. I’d say $1$ is the most reasonable candidate for “half-way between 0 and $\infty$”. Looking at it that way, neither $\pi$ nor $e$ is small at all – they’re actually way closer to $\infty$ than they are to 0 🙂

This is only a wild conjecture, but maybe one could consider the Kolmogorov complexity of mathematical constants. If we define a constant as a number that has many occurrences in entirely different contexts, then it’s obvious that the Kolomogorov complexity for a constant needs to be very low, so that it is likely its calculation specification re-emerges in many different places.

In fact, mathematical constants seem to have a very low Kolmogorov complexity, for example the constants on Wikipedia only require 15 mathematical symbols on average.

I have an intuition but not a proof that numbers with low Kolmogorov complexity either diverge or stay relatively small.

It’s ultimately a function of the Anthropic Principle. If things were different, we may not even be here. The universe needs to have certain relationships and entities and frameworks present in order to even be a universe at all, let alone a universe that is able to give rise to sentient, living creatures.

Some relationships in reality will be simple enough as to use meager, small-integer ratios. Others will be far more complex and wide-scale and may involve larger-numbered relationships. As for irrationality, I think that is a side-effect of not having a clear-cut, integral universe. When you start getting into frameworks that involve curves, you oftentimes need to invoke irrational numbers that encapsulate the concept of the infinite, which appears often in nature.

Actually the assumption behind your question depends on the status of the Riemann hypothesis. If there Riemann hypothesis were false, then the imaginary part of the first zero off the critical line would be a counterexample to your assumption.

I suppose if the Riemann hypothesis were undecidable, one could conjecture that so is your question. The equivalence of the two conjectures may be closely related to the status of the twin prime conjecture…