“A Handbook of Integer Sequences” Fifty Years Later

New paper by N. J. A. Sloane.

Abstract: Until 1973 there was no database of integer sequences. Someone coming across the sequence 1, 2, 4, 9, 21, 51, 127, . . . would have had no way of discovering that it had been studied since 1870 (today these are called the Motzkin numbers, and form entry A001006 in the database). Everything changed in 1973 with the publication of A Handbook of Integer Sequences, which listed 2372 entries. This report describes the fifty-year evolution of the database from the Handbook to its present form as The On-Line Encyclopedia of Integer Sequences (or OEIS), which contains 360,000 entries, receives a million visits a day, and has been cited 10,000 times, often with a comment saying “discovered thanks to the OEIS”.

I’m proud to have a couple of sequences in OEIS:

• A140961 (2008, with thanks to Vladeta Jovovic, whom I found via A051588, for the interpretation in terms of binary matrices). This arose when I was counting finite models of categorical syllogisms, thinking this might be useful for the psychology of reasoning. It wasn’t.
• A358693 (1 Jan 2023) – whilst looking for properties of the number 2023. Trivial extension of A001102.

Earliest Uses of Symbols of Set Theory and Logic

In case you too were wondering about the history of ⊢, ⊨, and friends, try this MacTutor page at the University of St Andrews, Scotland.

Kharkiv, statistics, and causal inference

As news comes in (14 May 2022) that Ukraine has won the battle of Kharkiv* and Russian troops are withdrawing, it may be of interest to know that a major figure in statistics and causal inference, Jerzy Neyman (1894-1981), trained as a mathematician there 1912-16. If you have ever used a confidence interval or conceptualised causal inference in terms of potential outcomes, then you owe him a debt of gratitude.

“[Neyman] was educated as a mathematician at the University of Kharkov*, 1912-16. After this he became a Lecturer at the Kharkov Institute of Technology with the title of Candidate. When speaking of these years he always stressed his debt to Sergei Bernstein, and his friendship with Otto Struve (later to meet him again in Berkeley). His thesis was entitled ‘Integral of Lebesgue’.” (Kendall et al., 1982)

* Харків (transliterated to Kharkiv) in Ukrainian, Харькoв (transliterated to Kharkov) in Russian.

Social Sciences under Attack in the UK (1981-1983)

Interesting paper by Michael Posner, who was chair of the UK Social Science Research Council (SSRC) when it was under attack by the Conservative Thatcher government in the early 1980s.

Secretary of State Sir Keith Joseph considered dismantling SSRC and asked for an independent review into its utility by an established biologist.

SSRC survived, though one notable change was made…

“Joseph opted for a public, but very light punishment: a change of name. I told him that I could persuade scores of academics to accept a name change if he would promise, on the record, the continuing independence of the SSRC. He agreed, and the SSRC was duly renamed the « Economic and Social Research Council » (ESRC). The significance of this change was the omission of the word « science », which Joseph had insisted upon and which many of us at the council and in academia found it difficult to accept.”

Ye olde Spearman

A good test of whether someone understands g is if they characterise it as a general factor in intelligence test scores and not as general intelligence (i.e., a substantive rather than statistical construct). But it’s interesting to see what Spearman originally said in his 1904 “General Intelligence,” Objectively Determined and Measured. On p. 272:

“… we reach the profoundly important conclusion that there really exists a something that we may provisionally term “General Sensory Discrimination” and similarly a “General Intelligence,” and further that the functional correspondence between these two is not appreciably less than absolute.”

He goes on to describe this as a “general theorem”, refining it to (p. 273):

Whenever branches of intellectual activity are at all dissimilar, then their correlations with one another appear wholly due to their being all variously saturated with some common fundamental Function (or group of Functions).”

(Enthusiastic emphasis in original.)

There’s a recent argument against this (though perhaps not quite, given Spearman’s parenthetical “group of Functions”), by van der Maas, et al. (2006). The abstract:

“Scores on cognitive tasks used in intelligence tests correlate positively with each other, i.e., they display a positive manifold of correlations. The positive manifold is often explained by positing a dominant latent variable, the g-factor, associated with a single quantitative cognitive or biological process or capacity. In this paper we propose a new explanation of the positive manifold based on a dynamical model, in which reciprocal causation or mutualism plays a central role. It is shown that the positive manifold emerges purely by positive beneficial interactions between cognitive processes during development. A single underlying g-factor plays no role in the model. The model offers explanations of important findings in intelligence research, such as the hierarchical factor structure of intelligence, the low predictability of intelligence from early childhood performance, the integration/differentiation effect, the increase in heritability of g, the Jensen effect, and is consistent with current explanations of the Flynn effect.”