What Does Big Mean?

What does it mean for a number to be big?

Not big-in-general. Big for a situation. An electricity bill can be $300 and that’s enormous to a college student and invisible to a hedge fund. The question isn’t the number. The question is whether the situation even registers the difference.

Your phone plan says unlimited but it isn’t actually infinite — there’s a finite number of bytes you could physically transfer in a month. But the cap is so far above your usage that hitting it is not a real possibility. Doubling it wouldn’t change your behavior. That’s what hyperfinite means. A specific quantity that’s large enough to behave like infinity for all purposes you care about. There IS a number. You just can’t reach it.

Big:

1. Your phone plan’s data cap. There’s a number. You’ll never hit it.
2. The number of molecules in the room you’re sitting in
3. The number of floating point operations your GPU runs during one training step
4. The pixel count of a satellite photo of your city at maximum resolution
5. The number of legal positions in Go

The pixels on your retina display are not infinitely small. They’re a specific finite size. But halving them — twice as fine — produces no perceptible change. The difference between the current pixel and the next smaller one is an infinitesimal for your visual system. The pixel isn’t zero. It’s just small enough that the difference between it and zero can’t be detected by the instrument, which is your eye.

Small:

1. The difference between your retina display and double its resolution
2. The thermal fluctuation of a single protein in your bloodstream
3. The gravitational effect of moving your coffee cup one inch to the left
4. The probability that every molecule of air in the room simultaneously moves to the left half
5. The error in GPS when you’re navigating between continents

A friend at Inkhaven is writing about fake numbers. His claim: a largest number exists, we just can’t know which one it is. Any number plus one would be larger, but at some point the numbers become so large they’re just — underspecified cognitive objects without real referent. He calls 2^2^2^100 a fake number. This is ultrafinitism.

I think he’s seeing something real and naming it wrong.

The feeling that 2^2^2^100 isn’t a real number — that’s not evidence that big numbers don’t exist. It’s evidence that big numbers are different from small ones in a way that ordinary math refuses to acknowledge. You can’t picture 2^2^2^100 any more than you can picture the Pacific Ocean, but the Pacific is there. The ultrafinitist says: if you can’t point to it, it’s fake. The nonstandard analyst says: you can point to it, it’s just infinite, and infinite numbers have different properties than finite ones. They don’t need to fit in your head. They need to be consistent.

The hyperreals contain numbers bigger than 2^2^2^100. They contain numbers bigger than any number you can name. But they also contain their reciprocals — infinitesimals smaller than anything you can measure. The ultrafinitist feels the bigness and concludes the number is fake. The nonstandard analyst feels the bigness and concludes the number is infinite. Same intuition, different ontology.

The transfer principle is why this works. Anything you can prove about finite numbers using first-order logic is also true about infinite ones. They’re not fake. They’re just big enough that doubling them doesn’t matter.

The universe itself is the best example.

The size of the Earth is roughly the geometric mean between the Milky Way and the nucleus of an atom — give or take three orders of magnitude. The ancient Greeks figured out the size of the Earth without knowing the size of the sun. Eratosthenes measured shadows in two cities and did some geometry. He didn’t need the distance to the sun because for his purposes it was infinite — the rays were parallel, which is what matters when the source is far enough away that it’s effectively at infinity.

Then you discover stars. Now you need a more refined notion of infinity, because the sun and Alpha Centauri are both “infinitely far” by Eratosthenes’ standard, but one is 4 light-years away and the other is 8 light-minutes. Regular infinity can’t tell them apart. Infinity squared is still just infinity. It loses information.

Hyperfinite numbers preserve the relative sizing. If H is an infinite hypernatural, then H, H², H³ are all infinite but they’re different infinities — H² is genuinely H times bigger than H. You can do arithmetic with them. You can divide them. You can compare them. The galaxy is H³ atoms wide, the sun is H² atoms wide, and you can actually compute the ratio and get H — the number of suns that fit across the galaxy. Regular infinity can’t do this. Infinity divided by infinity is undefined. H³ divided by H² is H.

This is the asymmetry that’s been bugging me. People have encountered infinitesimals throughout calculus — every DX, every epsilon, every limit. Infinite decimals are everywhere. But the dual concept — how many pieces is an integral split into — being an infinitely large natural number — is simply missing from the standard curriculum. Infinitesimals live in the continuous world. Hyperfinite numbers live in the discrete world. And discrete infinite quantities are useful in combinatorics, which is why even ultrafinitists and people who work in combinatorics appreciate them — you can do the counting arguments that you always wanted to do at infinity but couldn’t because infinity wasn’t a number. Now it is.

A continuous family of hyper discrete objects that all round off by infinitesimal error to a standard one. Like a circle and its infinite family of hyperfinite polygons. Or the normal distribution and its binomials.

“People have an intuition for infinitesimals, which is the real reason that people argue about whether 0.999… is one or not. It doesn’t matter how it’s defined. The point is — normie people — yes, you can keep dividing it, but no, just because you divide something, something can’t become nothing upon division.”

Here is the thing. Mathematicians had all this intuition and then threw it away.

For 150 years, from Leibniz to Cauchy, calculus was done with infinitesimals. Leibniz’s notation — \(\frac{dy}{dx}\), \(\int f\, dx\) — is infinitesimal notation. Then Weierstrass formalized calculus with epsilons and deltas. No infinitesimals, just limits. Clean. Rigorous. And also: you lost the picture.

What nonstandard analysis does is give you the intuition back as a theorem.

The move: a number is unlimited if it’s bigger than any real number. A number is infinitesimal if it’s smaller in absolute value than any positive real number. Formalize this with an actual number system — the hyperreals — and now your everyday intuitions about “operationally infinite” and “operationally zero” are rigorous mathematics.

“The advantage of this over the other conceptions of the infinite is that you can just use everything you already know. And isn’t that kind of appealing?”

The hyperreals contain all the real numbers, plus numbers bigger than all of them (unlimited), plus numbers crowded between 0 and every positive real (infinitesimals). Pick an unlimited number \(H\). Then \(H + 1\) is bigger by exactly one. \(H - H/3\) is smaller by exactly \(H/3\). \(\frac{1}{H}\) is infinitesimal. \(\frac{1}{H^2}\) is a smaller infinitesimal. They’re distinct. Infinitesimals have structure.

Unlike the usual \(\infty\), where \(\infty + 2.3 = \infty\) (throwing away information), \(H + 2.3\) is exactly \(2.3\) more than \(H\). A physicist would never mix up \(dx\) and \(dx^2\) — one is a line element, the other an area. Why treat infinity differently?

“The zeros of analysis, the soft zeros are really epsilons. I have a whole talk on how every case in analysis where zero shows up, it’s really an infinitesimal.”

There is a standard part function. It takes any hyperreal that’s close to a real number and rounds it to that real. \(\text{st}(5 + \frac{1}{H}) = 5\). The infinitesimal \(\frac{1}{H}\) rounds away. Unlimited numbers have no standard part — they’re not close to any real.

“Aspects as a good name for standard parts.”

From etymology: aspect is ad + spec, toward-looking. Taking the standard part is looking at something from far enough away that the infinitesimal detail disappears. You see the aspect — what it looks like from standard distance.

I have a post on big-O notation where this pays off concretely. Standard big-O says: \(f \in o(g)\) if for all \(\varepsilon > 0\) there exists \(x_0\) such that for all \(x > x_0\), \(\frac{f(x)}{g(x)} < \varepsilon\). Three quantifiers.

The NSA version: pick any unlimited \(H\) and look at \(\frac{ {}^*f(H)}{ {}^*g(H)}\) directly. \(f \in o(g)\) iff that ratio is infinitesimal.

“This is basically internalizing the definitions from big O. If you take a limit to infinity to look at the behavior of some function far out there. Well, you are far out there. In fact, you’re like infinitely far out there.”

Example: is \(x^2 \in o(x^3)\)? Evaluate at \(H\): \(\frac{H^2}{H^3} = \frac{1}{H}\), which is infinitesimal. Yes. Is \(x^3 \in O(x^2)\)? \(\frac{H^3}{H^2} = H\), which is unlimited. No. These are the same calculations you did in your head when you listed those 5 big quantities above — you were checking whether the ratio was infinite, zero, or bounded.

A function is continuous, in the nonstandard sense, if infinitely close inputs give infinitely close outputs: \(x \approx x' \implies f(x) \approx f(x')\). Continuous and bounded use the same definition — just with different equivalence relations. Continuity is the infinitely-close relation. Boundedness is the limited-distance relation. The same conceptual template, different thresholds.

I wrote about discontinuous linear maps: if \(H\) is unlimited, \(T(x) = Hx\) maps \(\frac{1}{H}\) and \(\frac{2}{H}\) — infinitely close — to \(1\) and \(2\) — not infinitely close. That’s the same failure mode as a very loud amplifier: it takes things that are close enough to be indistinguishable and blows them apart.

The transfer principle is what closes the loop. It says: any first-order statement true about the reals is true about the hyperreals, and vice versa.

Your intuition that some quantities are operationally infinite isn’t just useful — it’s provably correct as a hyperreal statement, and transfer pulls it back to the reals. The electricity bill is literally unlimited relative to the coffee. The ink on the mortgage is literally infinitesimal relative to the house price. These aren’t metaphors. They’re hyperreal inequalities.

“They’re more useful than both the cardinals and the ordinals.”

Cardinals and ordinals tell you the size of infinity — \(\aleph_0\) for countable, \(\aleph_1\) for the next size up. Useful. But you can’t do algebra with them. \(\infty + 1 = \infty\) collapses the structure. The hyperreals give you \(H + 1 > H\) and \(H + 1 < H + 2\), with all the usual arithmetic rules intact. That’s the difference between “the ship is really far away” and “the ship is 4.7 nautical miles northeast.”

“Learning infinitesimal hyperfinite numbers is literally like learning division without multiplication. Or learning division before multiplication, which is just perverse.”

This is directed at the standard curriculum. We teach limits before infinitesimals, even though infinitesimals came first and are what motivated the notation. Leibniz wrote \(dx\) and meant it. Weierstrass cleaned it up and in the process made it harder to think about. Robinson, in 1960, proved you could have both: rigorous and infinitesimals. We’ve just been teaching the harder version ever since.

One more thing.

“The statement is equivalent to the twin primes conjecture. Twin primes if and only if there exists one pair of primes, capital P, an infinite one and \(P + 2\).”

Standard phrasing of twin primes: for all \(n\), there exist primes \(p > n\) such that \(p\) and \(p + 2\) are both prime. NSA phrasing: there exists an unlimited prime \(P\) such that \(P + 2\) is also prime. Same conjecture. One requires a quantifier over all naturals. The other requires one number.

“A continuous object, any continuous object, like the circle, will be the standard part of an uncountable family because there’s loads of hyperfinite approximations.”

“You can approximate a continuous object by this sort of hyper discrete approximation. A hyperfinite number of infinitesimal pieces.”

This is the germ of the derivative-at-a-discontinuity post. A circle with \(H\) sides — \(H\) unlimited — is infinitely close to the actual circle at every point. The standard part of that polygon is the circle. Not an approximation in the colloquial sense. The actual thing, seen from standard distance.

What does big mean?

Big enough that making it bigger doesn’t change the situation.

That’s not fuzzy. It’s mathematics. The electricity bill is literally unlimited relative to the coffee. The ink on the mortgage is literally infinitesimal relative to the price. Pick your unlimited \(H\), build the hyperreals, invoke transfer. The number system you learned isn’t wrong. It’s just small.

Draft. Assembled by Claude from my talks and rants, conversations recorded by Limitless, and my existing blog posts on nonstandard analysis. The 30 examples are Claude’s. The math is mine. The part where I said “normie people” is definitely mine. I’m publishing this early so I can spend the rest of the day refining it — expect edits.

Credit to Abraham Robinson, who built this in 1960. Credit to Mikhail Katz and Jerome Keisler, who argue we should teach it this way.