Does the truth of a statement depend on how many people share it?

Imagine a room. Eight people sit in it. A line is shown to them, and each is asked to name aloud which of three other lines matches its length. Difference among the lines runs from three quarters of an inch to an inch and three quarters. One glance is enough to see it. Seven of the eight, one after another, give the same plainly wrong answer. They have arranged it beforehand. The eighth person’s turn arrives. It was this room, constructed by Solomon Asch in 1951, that measured something we feel but can rarely state as a number: what it costs a person to stand by their own judgment when the majority speaks otherwise.

Does a statement become true because of how many people share it? The answer that measurement yields is twofold. On its own, the number is a statistic about people. It turns into a statistic about the world under two conditions, sufficient within the Condorcet theorem: each person judges independently of every other, and each person’s probability of a correct answer exceeds one half. The Marquis de Condorcet formulated these conditions; Francis Galton saw them play out at a fair in Plymouth.

The argument from majority endorsement

The Internet Encyclopedia of Philosophy defines the appeal to the people (argumentum ad populum) through this instruction: “If you suggest too strongly that someone’s claim or argument is correct simply because it’s what most everyone believes, then your reasoning contains the Fallacy of Appeal to the People.” The reverse side of this fallacy is given by the encyclopedia in a second sentence: “Similarly, if you suggest too strongly that someone’s claim or argument is mistaken simply because it’s not what most everyone believes, then your reasoning also uses the fallacy.”

The key word is “not necessarily.” The Encyclopedia writes: “Agreement with popular opinion is not necessarily a reliable sign of truth, and deviation from popular opinion is not necessarily a reliable sign of error.”

Long before modern logic, Plato in Book Five of the Republic drew a boundary between knowledge and opinion. Knowledge, for Plato, belongs to the infallible; opinion belongs to the fallible. In Shorey’s translation, opinion appears as something described as “darker than knowledge but brighter than ignorance.”

How far the gap between a claim’s supporters and the fact can stretch is shown by a story often retold but seldom checked at its source. In 1931 the book Hundert Autoren gegen Einstein appeared. In reality it gathered texts from twenty-eight authors; overall the publication names 121 opponents to the special relativity theory, as a 2020 Skeptical Inquirer article reports. The famous reply attributed to Einstein, namely “one would be enough,” is not confirmed by any primary source. The article states: “Due to the lack of a primary source, it is unclear if Einstein made this kind of statement.” The phrase arrives through a secondary source, “Hawking (1988).” Whoever spoke it, the thought does not depend on authorship: against a fact, one hundred authors weigh no more than one.

When the crowd knows

At the annual Plymouth livestock show, visitors guessed on tickets how much a live ox would weigh after dressing. After the contest the tickets were “kindly lent me for examination,” Francis Galton wrote. He discarded thirteen faulty cards, and 787 guesses remained. According to his published data, the middlemost estimate was 1207 pounds, and the dressed ox weighed 1198 pounds. The error is 0.8 percent. Between 1162 and 1236 pounds fell the guesses’ middle half.

The statistician Kenneth Wallis checked Galton’s working tables in 2014 and found that “the outcome has been incorrectly transcribed.” The median is the value in the middle after sorting all estimates from smallest to largest. Wallis found it stood at 1208 lb. The true weight, from a letter by the organizer, was 1197 lb. The median’s error came to 11 lb., or 0.92 percent. The arithmetic mean, the sum of all estimates divided by their count, Wallis located in Galton’s own reply to a reader: it was 1197 lb. For the mean the error was zero.

Fig. 1 compares the ox’s true weight per Wallis’s corrected data against the median, the mean, and the guesses’ middle half. The range 1162–1236 lb. contains the true value of 1197; the median of 1208 misses it by under one percent; the mean of 1197 matches it exactly.

Galton described the guesses as “unbiassed by passion and uninfluenced by oratory and the like.” The theorem named in the next paragraph lists two conditions; these words describe the first of them: each visitor wrote down a guess without seeing the guesses of others. It was precisely this independence of judgments that let the crowd guess the ox’s weight almost without error.

Long before Galton, the Marquis de Condorcet formulated when a majority becomes evidence. His jury theorem rests on two assumptions. First: unconditional independence, that is, the events of individual judges giving a correct answer do not depend on one another. Second: the competence of each judge exceeds one half and is the same for all. As the group grows under these conditions, the probability grows and tends toward one that the majority is right.

Numbers make the logic concrete. At 55 percent probability of a correct answer per judge, one judge gives 55%; a group of 11 gives 63.3%; 101 gives 84.4%; 1001 gives 99.9%. If instead each person’s probability falls below one half, say 45 percent, a larger group works in reverse: 45% for one, 36.7% for 11, 15.6% for 101, 0.1% for 1001.

Fig. 2 compares the probability of a correct majority across group sizes at two levels of competence: 55% and 45%. Above one half, the majority of a large group is almost certain to be right. Below one half, the majority of a large group is almost certain to be wrong.

Every visitor at Plymouth wrote a guess on their ticket. In Asch’s room, the answers the others spoke aloud could be heard by each participant.

When the majority presses

In 1951 Solomon Asch set up an experiment that measured the cost of abandoning one’s own judgment. The participants were male students. Each was seated in a group of seven others who had been instructed beforehand to give the same incorrect answer on twelve of the eighteen trials. The task was simple: choose from three lines the one that matched the sample.

Among fifty critical subjects, 32 percent of all estimates given under pressure of a unanimous majority yielded to that majority. The remaining 68 percent stayed correct. Control groups who wrote their estimates privately showed “the virtual absence of errors.” One fourth of the critical subjects stayed completely independent. At the other extreme, one third displaced their estimates toward the majority in half or more of the trials.

A 1955 Scientific American article gave data for 123 participants. In private, errors occurred under one percent of the time. Under group pressure, 36.8 percent of judgments became erroneous. Roughly one quarter never went along with the wrong majority. Group size mattered up to a point: against a single opponent, answers remained independent in nearly all trials; two opponents pushed errors to 13.6 percent; three opponents brought 31.8 percent. Beyond three, pressure barely grew.

Fig. 3 compares the share of erroneous answers under different conditions: private written form (under 1 percent), two opponents (13.6 percent), three opponents (31.8 percent), a unanimous majority (36.8 percent), one ally present (5.5 percent), a 2023 replication (33 percent), and a replication paying for correctness (25 percent).

The figure “75 percent” that sometimes circulates in retellings needs a closer reading. In the 1955 article Asch wrote beneath a diagram: “Seventyfive per cent of experimental subjects agree with the majority in varying degrees.” This counts everyone who yielded at least once. The measured rate of erroneous answers is 36.8 percent. The one third in the 1951 article refers to those who yielded in half the trials or more.

The strongest objection: participants might have changed their public answer to dodge conflict while privately keeping their own judgment. Asch measured public answers; the data settle neither side of that question. Yet with one ally, the rate of yielding drops to 5.5 percent; with money for accuracy, errors drop from 33 to 25 percent. Both are measured changes in public answers.

With 210 subjects from the campus of the University of Bern, a replication by Franzen and Mader in 2023 yielded 33 percent errors on the classic task. When correct answers were paid for, errors dropped to 25 percent. When the task involved political statements instead of lines, the conformity rate reached 38 percent. A 1996 meta-analysis by Bond and Smith, covering 44 exact replications, gave an average error rate of 25 percent. Most of these replications were conducted with male students in the US. Asch’s result is confirmed by more recent studies in Japan, and also in Bosnia and Herzegovina. The effect is therefore neither American nor an artifact of the middle of the twentieth century.

When people only see one another

What happens to collective judgment when people see only other people’s estimates, with no demand to agree and no unanimous majority pressing on them?

Lorenz, Rauhut, Schweitzer, and Helbing in 2011 seated 144 participants, all ETH Zürich students, in separate booths and gave them six factual questions, each with five possible answers. Some could see the estimates of others after every round. Some never saw them. The finding is stated plainly: “knowledge about estimates of others narrows the diversity of opinions to such an extent that it undermines the wisdom of crowd effect in three different ways.” Diversity shrank “without improvements of its collective error”; the truth migrated to “peripheral regions of the range of estimates”; and after answers converged, participants grew more confident. People became more alike in their errors and more certain they were right.

Lorenz et al. also measured how the method of combining answers changes the outcome: the simple mean beat the first individual guess in only 21.3 percent of cases; the log-scale mean beat it in 77.1 percent.

A market where buyers see the choices of earlier buyers offers another measure of the same phenomenon. Salganik, Dodds, and Watts in 2006 recruited 14,341 participants, “recruited mostly from a teen-interest World Wide Web site,” to download unknown songs in one of nine parallel worlds. In eight worlds, a participant saw how many times each song had already been downloaded in that world. One world provided no such information. Result: “Increasing the strength of social influence increased both inequality and unpredictability of success.” Across the eight worlds with influence, success became more unequal than it was in the independent one. The authors sum it up: “The best songs rarely did poorly, and the worst rarely did well, but any other result was possible.” Popularity fed itself; what became a hit in one world could stay unnoticed in another.

In 2015, Bakshy, Messing, and Adamic analyzed 10.1 million Facebook users in the US. Their conclusion: “Compared with algorithmic ranking, individuals’ choices played a stronger role in limiting exposure to cross-cutting content.” The feed ranking cut the chance of seeing cross-cutting content by 5 percent for conservatives and 8 percent for liberals. People’s own clicks cut it by 17 and 6 percent.

Consensus that fell apart, and consensus imposed

Cases where observation or experiment opposed the prevailing view of their era, and one case where a view was enforced by power, appear in the historical record. Every one of these cases permits the collision to be interpreted via the two conditions identified earlier: independence of judgment and competence greater than one half.

Galileo Galilei was the first to report telescopic observations of mountains on the Moon, the satellites of Jupiter, and the phases of Venus. In June 1633 he was found guilty of “vehement suspicion of heresy” and sentenced to imprisonment, commuted to lifelong house arrest. The verdict depended entirely on authority, with no role for any observation that another could have replicated; consequently, those who concurred did so under that authority, and the first condition was missing.

In 1887 Albert Michelson and Edward Morley published a result they had not been looking for. They built an interferometer to measure Earth’s motion through the ether, the hypothetical medium then thought to carry light. The interference fringes were expected to shift in a way that revealed this motion, yet the observed shift was “certainly less than the twentieth part” of the prediction and “probably less than the fortieth part.” Earth’s speed relative to the ether was found to be “probably less than one sixth the earth’s orbital velocity, and certainly less than one-fourth.” Though the medium was the era’s shared presupposition, a single measurement counted more than the multitude who embraced it; competence rested on that assumption, and the measurement decided everything on its own.

In 1864 Louis Pasteur delivered a lecture summing up a series of experiments on spontaneous generation, the doctrine that life can spring from inanimate matter. His verdict was categorical: “Never will the doctrine of spontaneous generation recover from the mortal blow of this simple experiment.” One repeatable experiment settled the issue, and how many adhered to the doctrine contributed nothing whatsoever to that outcome.

The fourth case is different in nature: here the consensus was imposed by force. Nikolai Vavilov founded and headed the Lenin All-Union Academy of Agricultural Sciences in 1929. In 1935 he was removed from his post, and his place was taken by Trofim Lysenko. Vavilov died in prison. In August 1948, at a session of the academy, genetics was “fully defeated” and the state authorities “banned research” in the field. At the Institute of Plant Industry, at least ten leading researchers were arrested and either shot or died in prison; twelve spent years in imprisonment or exile. A letter against Lysenko was signed by over three hundred scientists in 1955. Khrushchev, however, “also supported Lysenko,” and genetics returned to the curriculum only after he lost power in 1964. A consensus maintained by force directly destroys the first condition, and the letter from over three hundred scientists is an independent judgment that resurfaced before the state permitted it.

The majority in price

A price on an exchange is another record of the majority’s view, in money rather than words. When the price climbs, the majority of money in the market votes that the asset is worth more. Three famous episodes survive, and their scale is visible in the numbers.

The economist Peter Garber revisited the episode of tulips in 17th-century Holland in 1990. A Semper Augustus bulb already cost 2,000 guilders in 1625, a decade before the well-known winter. The claim Garber examines says that in February 1637 the bulbs “could not be sold at 10 percent of their peak values.” Garber himself estimates that the decline “can have accounted for no more than a 16 percent price decline.” He also notes that the movement of prices for the bulbs was “typical of any market for rare bulbs.” Garber does not dispute the crash itself but bounds its size to what the price records show.

Based on Peter Garber’s data, a South Sea Company share had a value in January 1720 of about 120 pounds, reached roughly 775 pounds by August 31, and on October 1 fell to around 290 pounds. The peak was 6.46 times the starting price. The drop came to 62.6 percent. Setting the start to 100, the index hit 646 and fell to 242.

The NASDAQ Composite index, published by the Federal Reserve Bank of St. Louis, stood at 743.58 points early in 1995. On March 10, 2000, it reached 5048.62, 6.79 times higher. On October 9, 2002, it had sunk to 1114.11, a 77.9 percent drop. Scaled to 100 at the start, the index rose to 679 and slid to 150.

Fig. 4 compares the South Sea Company index of 1720 and the NASDAQ Composite of 1995–2002, both starting at 100. Both soar above sixfold (646 and 679) and both fall back to 242 and 150.

Why don’t informed traders stop a bubble? Markus Brunnermeier, in an entry for The New Palgrave Dictionary of Economics, puts it as a synchronization problem: “a single trader alone cannot typically bring the market down by himself, coordination among rational traders is required and a synchronization problem arises.”

Consequences of these findings, and the limitations of the measurement

The study’s data make it possible to formulate when the majority’s opinion is evidence, not merely a statistic about the majority itself.

First, independence. At Galton’s fair, each visitor wrote down a guess on a ticket, without seeing the guesses of others. In Asch’s room, independence vanished: the subject heard the answers of a unanimous majority and yielded to it. Lorenz’s booths showed the same effect without pressure, merely from observing the estimates of others, and the song market showed it in a situation where social influence amplified inequality and unpredictability of success.

Second, the competence of those judging exceeds one half. The Condorcet theorem proves this mathematically: When each person’s chance of a right response is below one half, a smaller group errs less frequently than a larger one.

Third, what follows from the data concerns the behavior of the person standing before the majority. Asch’s measurements show: when a person writes down their judgment in private, errors are practically absent. When at least one ally appears in a room with a unanimous majority, yielding drops sharply. When correctness is paid for, errors decrease.

This study’s limits are set by what was measured and what was not.

• Asch’s participants were male university students; the 2023 replication used persons from the campus of the University of Bern. The findings do not cover other demographic groups, though the basic effect is confirmed by studies in Japan and Bosnia and Herzegovina.
• Lorenz’s participants were ETH Zürich students; Salganik’s were mostly visitors of a teen-interest website. Conclusions about the effect of seeing others’ estimates apply to these samples.
• Bakshy et al. cover one platform, Facebook, over one period.
• The Condorcet theorem is a mathematical model; real groups violate its assumptions to varying degrees.
• The four historical cases serve as illustration. No dataset measures what share of past consensuses later proved wrong.
• Garber’s “no more than a 16 percent” price decline for tulips is an upper bound he draws from price records.
• Every case considered involved statements for which a measurable fact exists: an ox’s weight, a line’s length, the phases of Venus, interference fringes, a stock price. For questions of taste, values, or law, this study provides no such measurable fact.

Galton showed that an independent crowd can be strikingly accurate. Asch showed that the crowd loses this accuracy as soon as its members begin hearing one another. Lorenz showed that pressure is not even required: knowing others’ answers is enough. The mean of Galton’s independent guesses gave zero error.