Multipliers and Limits: Doubling the Square Again

Plato's doubled square and a modern subscription with two limits show why a multiplier on one quantity rarely multiplies the quantity you care about.

A multiplier is attached to one quantity. The quantity you care about may be a different one, governed by a different rule. Plato saw the first half of this in a drawn square, and a modern subscription with two usage limits shows the second half. The geometry is Demonstrated. What the plans do is Provisional: the company's help articles give the multipliers but no figure for the weekly limit, and the terms can change.

The boy doubles the side

In the Meno, Socrates draws a square and asks a boy for a figure "twice the size of this, but of the same sort" (82d, Lamb's translation). The boy answers at once: "Clearly, Socrates, double" (82e). Double the side, he means. Socrates draws it, and the boy sees that "from the double-sized line" comes a space "not of double, but of fourfold size" (83c). He tries a side of three feet and gets nine, where eight was wanted (83e). Then he says, "I for one do not know" (84a).

Socrates counts that admission as progress. The boy "feels the difficulty he is in" and is "better off in respect of the matter which he did not know" (84a-84b). These quotations are Attested, from Lamb's translation. The answer, when it comes, is not a better guess at the side. It is a new construction: the square built on the diagonal (85b), which Elements I.47 later explains.

The boy's mistake was to assume that area moves in step with the side. It does not. Area goes with the square of the side, so doubling one quadruples the other. Two quantities, one rule linking them, and the rule is not proportion.

Two limits, not one

Anthropic sells its Max plan in two tiers, named for how much more they give than its Pro plan. In the company's words, "Max 5x includes five times the Pro plan's per-session usage allowance" and "Max 20x includes 20 times". The session limit "will reset every five hours". The same article adds that "Max plans also have a weekly usage limit that applies across all models", and that the company may impose "weekly and monthly caps or model and feature usage, at our discretion" ("What is the Max plan?", support.claude.com, read 9 October 2026). A second article says the limits "are shared across Claude and Claude Code" ("Use Claude Code with your Pro or Max plan").

So the larger tier is four times the smaller per five-hour window. The tempting conclusion is that it gives four times as much. That is the boy's first answer again. The multiplier is stated for one quantity, the window. The question most buyers care about is how much they can do in a week or a month, which is a different quantity, and a second rule governs it.

Height, width and area

Picture a week as a strip. Its width is time: 168 hours, about 34 five-hour windows. Its height is how much you use in one window, and the tier caps that height at 5 or 20 units. The area, height times width, is your use for the week, and the weekly limit caps the area.

The help articles read for this page give no weekly figure, so the numbers below are illustrations, not facts. Suppose the smaller tier's weekly limit is 50 units and the larger tier's is 100, a doubling.

Height (per window). Smaller tier: 5. Larger tier: 20. Ratio: 4.
Box (height × 34 windows). Smaller tier: 170. Larger tier: 680. Ratio: 4.
Area allowed (weekly limit, illustrative). Smaller tier: 50. Larger tier: 100. Ratio: 2.
Full windows before the weekly limit. Smaller tier: 10. Larger tier: 5. Ratio: 0.5.

The multiplier stretches the height. The weekly limit caps the area. Height grows four times while the area only doubles, so the larger tier fills its weekly limit in half as many full windows: fewer, taller bursts.

Here the analogy with the Meno breaks, and the break is instructive. In the square, area follows from the side by necessity. In the plans, area does not follow from height at all. A second boundary, set by the seller on its own terms, decides it. The boy's error was not knowing the rule that links side and area. The buyer's error is assuming there is such a rule.

The ratio depends on the shape of your use

How much more the larger tier gives you depends on which boundary you meet first, and that depends on how you work. Two users, with the same illustrative limits:

A steady user uses 3 units in each of 15 windows a week, 45 in all. That fits under both heights and both weekly limits. The larger tier gives this user nothing extra: a ratio of 1.
A burst user wants 20 units in each of 4 windows, 80 in all. The smaller tier caps each window at 5. The larger tier allows all 80. Within those 4 windows the ratio is 80 to 20, or 4. If the work can be spread over more windows, the smaller tier still stops at its weekly 50, and the ratio over the week is 80 to 50, or 1.6.

The real ratio lies between 1 and the window multiplier, and your own pattern of use decides where. If you use everything both tiers allow, the weekly limits decide it, and the ratio is the weekly ratio: 2 in the illustration. This is the point made in Quantitative Reasoning about averages hiding bursts, seen from the other side: a limit on the average (the week) and a limit on the burst (the window) are different limits, and only the burst limit carries the advertised multiplier. A network engineer will recognise the same pair in a traffic shaper with a monthly data cap.

What uses up a window matters as well. Anthropic names "the length and complexity of your conversations", the features used, the model and the effort level, and adds that "longer conversations that trigger automatic context management consume more of your usage limit" ("How do usage and length limits work?"). Those are levers in the user's hands under any tier.

Where this could be wrong

The strongest objection is that the analogy is loose. The Meno is about necessity in geometry, and a pricing policy is a business decision. The reply is that the page claims only a shared shape of error: a number set on one quantity is carried over to another. The geometry teaches the habit of asking which quantity a number belongs to. It does not predict the price list. What would settle the practical question is the company publishing its weekly limits, or a user's own record of which limit stopped them and when. The illustrative 50 and 100 above could be far from the real figures in either direction.

Try this

Find a multiplier in your own work, such as a link "twice as fast" or a disk "four times larger". Name the quantity it applies to, then name the quantity you care about.
Find the second limit. For a faster link that might be the disk at the other end. For an AI plan it is the weekly limit; Settings > Usage shows when it next resets.
Draw your use as a strip of windows for one week, height for intensity. Mark where each limit would have stopped you.
Compute your own ratio between the two tiers from that drawing, and say which boundary decided it.
Read Meno 82b-85b and redo the boy's two wrong attempts with your own squares. Then build the square on the diagonal and check its area.

Related: Proof and Precise Reasoning: From Arguments to Theorems for why the boy's admission counts as progress, Geometry, Euclid and Trigonometry for the diagonal and Elements I.47, Building Bounded Agents for cost per useful result, and the Academy's elenchus for the method Socrates uses.