bits.

FREE CONTRIBUTION CHRONOLOGY · 1847–1890

Logic and people become data

What is gained, and what is erased, when reasoning or a person becomes a symbolic record?

The historical question

Boolean algebra made logical form calculable. Hollerith cards turned selected attributes of people into sortable fields and electrical counts, making large institutions faster and more capable.

A representation increases operational power by preserving selected distinctions and deleting the rest; the deletion is a technical and political choice.

An algebra and a data schema

Let A mean a record is in region A and E mean it meets an eligibility rule. Counting eligible records in either region A or B requires the predicate E AND (A OR B). Boolean algebra distinguishes that from (E AND A) OR B, which would include every record in B regardless of eligibility. Parentheses express a substantive rule, not decoration.

Now suppose a card has one field for age band and another for occupation. A tabulator can count combinations represented by those fields. It cannot recover information that was never encoded. If age was grouped into bands, exact ages cannot be reconstructed from band totals. If the occupation categories exclude a kind of work, a technically accurate count can still misrepresent the population.

This connects logic and institutional data without collapsing them into one invention. Algebra helps specify the counting predicate. Data design determines what distinctions exist. Collection procedures determine whether the records correspond to people accurately. Governance determines who chooses the categories and who can contest errors. A faster counting machine improves one stage of this pipeline; it does not prove the categories are appropriate or that the records are unbiased.

What the evidence does not establish

Historical accounts of data processing and state power need careful source comparison. Do not make either celebratory or accusatory claims broader than the evidence.

Reconstruction exercise

A dataset contains 10 eligible records in A,8 in B, and 3 in both. Count eligible records in A or B. Name information lost if ages are stored only as bands.

Worked solution

Inclusion-exclusion gives 10 + 8 − 3 = 15 unique eligible records. Adding 18 double-counts the intersection. Age bands lose exact values within each band, so an exact median age may not be identifiable. A downstream query must respect both set semantics and the precision of the stored representation.

Test the limits and transfer the idea

1. Truth table and counterexample · allow 25 minutes

A misplaced parenthesis changes who is counted

Compare P=E AND (A OR B) with Q=(E AND A) OR B. Evaluate both for records (E,A,B)=(0,0,1), (1,0,1), (1,1,0), and (0,1,0). Characterize exactly the records Q admits that P excludes. Then explain why a correct implementation of Q cannot be called accurate merely because all its counters match a hand calculation.

Get a starting hint

Find the branch where B alone decides Q, regardless of eligibility.

Compare your reasoning with the worked solution

The P,Q results are (0,1), (1,1), (1,1), and (0,0). Q differs precisely when B is true and E is false, independent of A. It admits ineligible B records because OR B bypasses the eligibility test. Matching hand counts shows that implementation and hand calculation agree on Q; it does not establish that Q is the intended institutional rule. Requirements should name eligibility, region overlap and how missing values are handled, with examples that distinguish the predicates. Boole's text is a primary logical work; the Census Bureau's history is a later institutional account. Neither alone determines a particular census classification's intended meaning or fairness.

Check your reasoning

  • Evaluates all four records correctly.
  • Characterizes the full difference, not only one counterexample.
  • Separates specification correctness from implementation agreement.

2. Information-loss argument · allow 25 minutes

Prove that a lossy category cannot answer a later query

A dataset retains only that four people have ages in the inclusive band 20–29. A later analyst asks how many are at least 25. Give two underlying populations with the same stored record and different correct answers. Derive the tight minimum and maximum possible count. Explain what additional data would be needed and why a faster tabulator cannot recover it.

Get a starting hint

Two inputs with identical encodings force any deterministic decoder to give the same answer for both.

Compare your reasoning with the worked solution

Ages [20,21,22,23] and [26,27,28,29] both encode as four records in band 20–29, but the requested counts are zero and four. Those are the tight bounds because no count can be negative or exceed the population of four, and both extremes are feasible. Any algorithm seeing only the stored record receives identical input in both cases, so it cannot always output the correct exact count. Exact ages or a retained sub-band boundary at 25 would answer this query, subject to collection accuracy and appropriate access. Additional computing speed cannot undo a non-injective encoding. Historical claims about population surveillance require source-specific evidence of categories, access and use; the mathematical loss argument alone does not establish those institutional facts.

Check your reasoning

  • Constructs indistinguishable encodings with different answers.
  • Derives and attains bounds zero through four.
  • Explains why the limitation is informational rather than computational.

Check your understanding

Why does 10 + 8 overcount the union?

  • Some records belong to both sets
  • Boolean algebra forbids addition
  • The categories must contain equal numbers
Explanation

Inclusion-exclusion gives 10 + 8 − 3 =15 unique eligible records. Adding 18 double-counts the intersection. Age bands lose exact values within each band, so an exact median age may not be identifiable. A downstream query must respect both set semantics and the precision of the stored representation.

Sources and further reading