Hi-Story Time x 2!

It’s story time! (Rather, hi-story time!) Here is a retelling of two truly terrific tales from the history of logic. I hope you find them fun and engaging!

  • Story Time 1/2: Russell’s letter to Frege announcing the contradiction

On 16 June 1902, Bertrand Russell (1872-1970) wrote to Gottlob Frege (1848-1925). Russell had just discovered his famous paradox, and he wanted to inform Frege that an analogue was derivable in the logic of Grundgesetze (Basic Laws of Arithmetic, link). Meanwhile, Frege had just sent to press the second volume of Grundgesetze, his life’s work. Then Frege got the news of Russell’s contradiction.

Just to pump your sympathy for Frege: imagine you had just finished and sent off your book or dissertation, you were days from your publication or your defense, and then — somebody points out you contradicted yourself on some point that was fundamental to your argument. I reckon I would be pretty upset, and I reckon you would be, too.

How did Frege react? Like this:

“Dear colleague: Many thanks for your interesting letter of 16 June…Your discovery of the contradiction caused me the greatest surprise, and, I would almost say, consternation, since it has shaken the foundation on which I intended to build arithmetic…In any case your discovery is very remarkable and will perhaps result in a great advance in logic, unwelcome as it may seem at first glance.”

That is a powerful response to what must have been a monumental disappointment. And Russell thought as much, too. In a 23 November 1962 letter reflecting on the exchange about 50 years later, Russell commends Frege’s response to the contradiction:

“As I think about acts of integrity and grace, I realise that there is nothing in my knowledge to compare with Frege’s dedication to truth. His entire life’s work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his fundamental assumption was in error, he responded with intellectual pleasure clearly submerging any feelings of personal disappointment. It was almost superhuman and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known.”

Frege’s prediction was right, of course: the contradiction led Russell to collaborate with Whitehead to produce Principia Mathematica. And Principia was a direct inspiration of enormously fruitful work, including Gödel’s theorems, Church’s work on type theory, and computer scientists’ research on artificial intelligence and automated theorem-proving in the 1950s and 1960s (including Newell, Simon, and Shaw’s Logic Theorist). Speaking of Principia:

  • Story Time 2/2: Whitehead and Russell had to self-publish Principia Mathematica

Imagine you spent ten years — say, your entire 30s — reworking and fixing the very foundations of mathematics and logic. You invent type theory, solve the contradictions, and put mathematics back on a firm foundations. You also prove a ton of interesting theorems about cardinals and ordinals just for kicks. An academic publisher would beg to print that work, right?

Wrong! It turns out Cambridge University Press estimated they would lose £600 in publishing Principia. (That amounts to almost $100,000 in today’s dollars.) So Cambridge University Press told Whitehead and Russell they would assume half that risk. That meant Whitehead and Russell, after working for ten years to fix math for everyone else, had to scrounge about for £300 to tell everyone what they did to fix it.

Happily, they got a grant for £200. But Whitehead and Russell each had to fork over £50 (~$8,000 today) to self-publish their grand treatise on the foundations of mathematics. And Cambridge thought they would lose money: probably a copy now resides in every college and university library in the world.

A little more detail on the calculations from 1910 UK pounds to modern US dollars are given here (link). And you should definitely read Principia (link) — yes, all three volumes (or at least skip the introduction and skim **1-20, pp. 90-199). Enjoy!

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