Unit 16 activity: The historical bug and historical honesty

Read this after attempting the activity.

This key does not restate the value asked for at cp16-note-g. Deriving it is the learning objective, and the checkpoint gives targeted diagnostics if you go astray. Checkpoint answers live in course/lab/checkpoints.py and nowhere else, by design.

Citation shorthand. (Menabrea 1843) = Menabrea, "Sketch of the Analytical Engine invented by Charles Babbage", trans. with Notes A–G by Ada Lovelace, in Taylor (ed.), Scientific Memoirs, vol. 3, 1843. Full entry in course/references.md.

Task 1 — the three written questions

1. What did the comparison actually catch?

A value-by-value comparison against a reference you obtained independently catches exactly one class of thing: a disagreement in a final value at a named index. That covers a wrong number, a wrong sign, an inexact number where an exact one was required, and — if you are comparing index against index rather than scanning for familiar-looking values — a value that is itself correct but filed under the wrong label.

That last case is worth dwelling on, because it is the one you met at cp08-diagnose. Every individual value can be a genuine Bernoulli number while the output as a whole is still wrong.

A good answer names the class of error, not just the outcome. "It caught that the numbers differ" is thin. "It caught a disagreement at a specific index, which is the only thing comparing final values can catch" is the answer.

2. What would it not have caught?

Several things, and any one of them earns the mark:

• A right answer reached by a wrong route. Two errors that cancel, or an operation that is wrong but happens not to affect this particular example, leave the final value intact.
• Errors elsewhere in the procedure. A single worked example exercises a single path. Anything the example does not reach is untested.
• Errors in the surrounding prose. A comparison of numbers says nothing about the claims made in the text around the table.
• The location of the fault. This is the important one. A final-value comparison tells you that something disagrees. It does not tell you which step is responsible.

That last point is the hinge of the whole unit. The value comparison supports the claim that the published table contains an error — a claim about final values. It cannot, on its own, underwrite the claim about where the error is. That location claim — operation 4, operands inverted — took a different kind of check: reading the facsimile of the 1843 printing [P2a] and recomputing the table's operations in noteg.py (see course/historical-notes.md §4). Each kind of claim needs its own kind of check, and the location claim moved to established only when that check was done. If you connected those two things unprompted, you have understood the unit.

3. What did you have to know before the comparison meant anything?

At least three things, and the third is the one the question is fishing for.

• The convention. Under the second Bernoulli convention, B1 = +1/2, one value in the sequence differs. Comparing against a reference built on a different convention produces a fake discrepancy.
• That your reference is independent. A reference derived from the thing you are checking checks nothing.
• What "B7" means in the source's own notation. This is not a fact about mathematics at all. It is a fact about how a document written in 1843 labels its quantities — Lovelace numbers only the non-zero Bernoulli numbers (Menabrea 1843), so her labels do not mean what the identical modern labels mean.

Reading a historical source requires knowing its conventions, and the conventions are usually not stated in the source, because at the time they did not need to be. Exactly the same is true of a specification handed to a generating system: the requirement most likely to be violated is the one so obvious to you that you did not write it down.

Task 2 — the seven statements

The bucket assignments below are this module's positions, stated as positions. If you placed a statement differently but gave a defensible reason and a concrete "what would move it", you have done the exercise correctly. The reasoning is the part that counts, not the label.

A. Note G appeared in 1843, with Lovelace's Notes appended

Established. Covered directly by the primary source (Menabrea 1843), and recorded in course/references.md.

What would move it: essentially nothing short of a bibliographic error in the volume itself. This is about as settled as a historical claim gets, and it is useful in the exercise precisely as a calibration point — the bucket is not empty.

B. The published table contains at least one error

Established. This module states it, and course/historical-notes.md gives the fuller detail.

Note how narrow the statement is. It asserts existence and nothing else — no location, no kind, no author. Marking this "contested" confuses it with statement C, which is the confusion the whole unit is built around — a productive mistake, and one worth sitting with rather than simply correcting.

What would move it: a careful published analysis of the original printing concluding that the table is in fact correct as printed.

C. The error is an inverted divisor and dividend in one operation

Established — and it is worth knowing that it only recently was.

For most of this module's development the right answer here was contested: the description is the one you meet most often, and wide circulation is not evidence, because many retellings may descend from a single earlier account rather than from the printing itself.

Then the check happened. The "what would move it" for this claim was always specific — a facsimile of the 1843 printing, read directly, with a trace of the operations against a correct computation — and it was carried out. The facsimile [P2a] shows the error in operation 4, acting on V₅ and V₄ in that order, where producing the stated (2n−1)/(2n+1) needs V₄ ÷ V₅. The operands are inverted. See course/historical-notes.md §4 and the unit reading, section 3.

So this claim moved buckets — not because it was repeated more, but because the consultable thing got consulted. If you marked this "contested", you gave the answer that was correct on the older evidence; section 3 is where the evidence changed. The reasoning counts either way, and the tell is whether it rests on the facsimile check or on the claim's popularity. What stays contested is who introduced it — items D and E — because no facsimile can show that.

D. Lovelace introduced the error; Babbage did not catch it

Contested. An attribution, and one of several in circulation. Attribution is a strictly harder claim than existence: it requires evidence about the process of production, not just about the printed result.

What would move it: surviving manuscript drafts, proof sheets, or correspondence from the period showing the value at different stages (unsourced; what manuscript material exists, and where it is held, is recorded as an open question in course/historical-notes.md).

E. The error was introduced by the printer

Contested, and for the same reason as D. It is a competing attribution, and the same evidence would bear on both.

If you noticed that D and E cannot both be right, and that this alone is enough to place both in "contested" without knowing anything else, that is a genuinely good observation. Mutual incompatibility among widely repeated accounts is a cheap and reliable contestedness detector. It costs nothing to apply and it works on generated text.

F. The error stood uncorrected for more than a century

Unsourced. This module has not verified when the error was first remarked on in print, or by whom (unsourced; recorded as an open question in course/historical-notes.md).

"Unsourced" rather than "contested" is the right call here because the module is not aware of competing accounts — it is simply aware of no checked account at all. "Contested" is a reasonable answer here; "established" is not, and the tell is that the statement is quantitative ("more than a century") while nothing supporting the number has been produced.

What would move it: a dated published correction, or a survey of the literature identifying the earliest one.

G. The as-printed table computes −25621/630

Unsourced as stated — and it stays unsourced after checking, because the statement does not say what it is about.

Before any check, this is a textbook bucket-two-or-three claim: the figure is everywhere online and cited to no scholarly source, and repetition is not corroboration. Unlike D and E it looks checkable from your desk, and this module checked it — and got the check wrong in an instructive way.

Note G's table has more than one printed fault, and it consumes earlier Bernoulli numbers that may be supplied or generated. Running python3 noteg.py in course/lab/ prints what each reading produces: −1/30 intended, 139/630 with operation 4 as printed and operation 24 repaired, −139/630 with both faults literal, and −25621/630 when the faulty routine generates the earlier values it then consumes. The last of those is the repeated figure, reproduced exactly.

So the honest verdict is not "refuted". It is that the phrase "the as-printed table" does not pick out one object, so the claim cannot be true or false until its interpretation is stated. An earlier version of this module marked it refuted on the strength of a run that had silently repaired operation 24 — the same failure, committed by the people warning about it.

Filing this under "established" because the number is widely repeated is the misfiling the unit warns about. Filing it under "refuted" because a recomputation disagreed is the subtler one.

What would move it: stating the interpretation. "Operation 4 as printed, operation 24 repaired, earlier values supplied" is a claim that can be checked, and checks out at 139/630. "The routine run from the start with the operation-4 inversion in place" checks out at −25621/630. A high-resolution facsimile confirming operation 24 independently of [S7] would firm up the whole family.

Common ways this activity goes wrong

Treating "contested" as a polite way of saying "false". It is not. It is a statement about the strength of the evidence available, not about the truth of the claim. Statement C may well turn out to be an accurate description; the module's position is that it has not established this, not that it doubts it.

Sorting by plausibility instead of by provenance. The buckets are about where a claim came from, not about how likely it sounds. A highly plausible claim with no source is unsourced. This is the single most common error in the task.

Producing a vague "what would move it". "More research" and "checking the sources" are not answers. The test is whether you have named something a person could actually go and consult. If you cannot, you have not yet worked out what the claim depends on.

Looking it up before sorting it. Understandable, and it destroys the exercise. The skill being trained is what you do with a claim before you have resolved it, because in practice most claims are never resolved — they are used, or not used, in whatever state you found them.

Why this unit is written the way it is

• It is deliberately short and deliberately under-coloured. If it feels thin next to the rest of the module, that is the design: this is the unit where invented historical detail is most likely to creep in, from any author, human or otherwise. Restraint here is the same discipline the unit is teaching.
• The question worth carrying out of it is not "which bucket" but "what would move it". That one generalises to essentially every claim you will meet in the rest of your degree.
• If you found the sorting easy, statement F is the one to sit with. It is quantitative, memorable, entirely uncontroversial in tone, and unsupported — which describes a large fraction of confident writing.
• Everything historical here defers to course/historical-notes.md. If the evidence changes, that file changes first; this unit deliberately holds no independent historical position that could drift out of step with it.