Engines of Our Ingenuity

No. 2805: GOTTLOB FREGE

Andrew Boyd

Today, a man of logic. The University of Houston's College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

Fans of Star Trek's Mr. Spock may have walked away with the impression that "logical" was the opposite of "emotional." But logic is something far different. It's about what makes a valid argument and what's simply nonsense.

Frege portrait

All men are mortal. Socrates is a man. Therefore Socrates is mortal. It's a valid argument. We can argue about whether all men are mortal, and quibble about whether Socrates is a man. But if we agree these two premises are true, logic dictates we must conclude Socrates is mortal.

The history of logic is long and involved. For centuries scholars grappled with the right way to capture and express its rules. But that changed in the late nineteenth century thanks to the work of Gottlob Frege.

Frege was trained as a mathematician, well versed in its formal symbols and rules; symbols like the plus sign and rules like 3 plus 4 equals 4 plus 3. So Frege set out to systematize logic by building his own formal symbols and rules.

logic expression

In 1879 he published a landmark work entitled Concept Writing: A Formal Language for Pure Thought Modeled on that of Arithmetic. For years it received little attention, but it would later be hailed as a monumental turning point in the development of logic.

Frege believed so deeply in his formal logic that he turned the tables and set about to define arithmetic in terms of logic. He worked for over ten years on a two volume work entitled the Basic Laws of Arithmetic. But just as the second volume was going to press, he received a letter from Bertrand Russell pointing out a fundamental error. Frege immediately understood the problem and added a last minute appendix where he wrote, "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished." We can only imagine the overwhelming despair he must have felt. Frege compounded his embarrassment when his proposed remedy proved equally ill-fated.

Bertrand Russell

Mistakes of this order usually define a scholar's legacy. Thankfully, Frege is less remembered for his missteps than for the pioneering nature of his work. Heavyweights Russell and Ludwig Wittgenstein went on to praise and promote Frege's contributions. And Frege's systematic approach to logic helped set in motion a wave of research. Some of the greatest mathematical minds of the twentieth century turned their attention to fascinating new problems in logic, including David Hilbert, John von Neumann, and Kurt Gödel. But perhaps Frege's greatest legacy is that scholars now generally agree mathematics is built upon a framework of formal logic. Frege may have had the details wrong, but he was spot on when it came to the bigger picture.

I'm Andy Boyd at the University of Houston, where we're interested in the way inventive minds work.

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Notes and references:

The error communicated by Russell to Frege is the now famous Russell's Paradox, describing a set whose definition is self-contradictory, but definable in Frege's system. The set is "the set containing all sets that don't contain themselves as members." The set can't contain itself by definition, but because it doesn't contain itself, it does. For more on the paradox, see the reference "What is Russell's Paradox?" below.

Gottlob Frege. From the Stanford Encyclopedia of Philosophy website: http://plato.stanford.edu/entries/frege/. Accessed June 26, 2012.

Gottlob Frege. From the Wikipedia website: http://en.wikipedia.org/wiki/Frege. Accessed June 26, 2012.

What is Russell's Paradox? From the Scientific American website: http://www.scientificamerican.com/article.cfm?id=what-is-russells-paradox. Accessed June 26, 2012.

The pictures of Frege and Russell are from Wikimedia Commons.

This episode was first aired on June 28, 2012



The Engines of Our Ingenuity is Copyright © 1988-2012 by John H. Lienhard.