{"id":97905,"date":"2022-12-15T14:15:21","date_gmt":"2022-12-15T21:15:21","guid":{"rendered":"https:\/\/www.patheos.com\/blogs\/danpeterson\/?p=97905"},"modified":"2022-12-16T20:32:40","modified_gmt":"2022-12-17T03:32:40","slug":"are-logic-and-mathematics-purely-rational-2","status":"publish","type":"post","link":"https:\/\/www.patheos.com\/blogs\/danpeterson\/2022\/12\/are-logic-and-mathematics-purely-rational-2.html","title":{"rendered":"Are logic and mathematics purely rational? (2)"},"content":{"rendered":"\n<\/head>

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The Rosette Nebula (Wikimedia Commons public domain image)<\/figcaption><\/figure>\n

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Yesterday, I posted a brief preliminary note inspired by and drawing upon Michael Guillen\u2019s Believing is Seeing: A Physicist Explains How Science Shattered His Atheism and Revealed the Necessity of Faith<\/em> (Carol Stream, IL: Tyndale Refresh, 2021).\u00a0 It concerned the modern collapse of confidence in the \u201ccertainty\u201d of mathematics and logic, which have surely seemed the most certain of all human intellectual pursuits.\u00a0 But I didn\u2019t finish my note, and so I continue it here.<\/p>\n

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This is Michael Guillen himself speaking, summarizing the implications of Kurt G\u00f6del\u2019s famous (infamous?) \u201cincompleteness theorem,\u201d which was mentioned in yesterday\u2019s blog entry:<\/p>\n

Whenever you try thinking logically about a complicated subject, one of two things will always happen:<\/span><\/p>\n

Possibility #1:<\/strong>\u00a0 You\u2019ll say and believe something that\u2019s genuinely true, but you\u2019ll never be able to prove<\/em> it.\u00a0 No matter how hard you try, logic will fail you because logic is not powerful enough to do the job.<\/span><\/p>\n

The late Swiss-American logician Verena Huber-Dyson put it this way:\u00a0 \u201cThere is more to truth<\/em> than can be caught by proof.<\/em>\u201d\u00a0 I prefer simply saying, Truth is bigger than proof<\/em>.<\/span><\/p>\n

Possibility #2:<\/strong>\u00a0 You\u2019ll prove<\/em> something is true using seemingly air-tight logic; but in fact, it\u2019s not so.\u00a0 Even though your logic seems rigorous, it isn\u2019t; it\u2019s riddled with stealthy paradoxes.<\/span><\/p>\n

As Morris Kline, the renowned American mathematician, wrote in his textbook Mathematics for the Nonmathematician<\/em>, \u201cLogic is the art of going wrong with confidence.\u201d\u00a0 (112)<\/span><\/p><\/blockquote>\n

Bertrand Russell had spotted a lethal logical error in the first pages of Gottlob Frege\u2019s massive Grundgesetze der Arithmetik<\/em> and, now, Kurt G\u00f6del had identified an even more fundamental error, not only in Bertrand Russell\u2019s (and Alfred North Whitehead\u2019s) similarly ambitious Principia Mathematica<\/em> but at the foundation of mathematics and logic themselves.<\/p>\n

\u201cI wanted certainty in the kind of way in which people want religious faith,\u201d the vocally and famously atheistic Russell mourned late in his life in Portraits from Memory<\/em>.\u00a0 \u201cI thought that certainty is more likely to be found in mathematics than elsewhere.\u00a0 But . . . after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I<\/em> could do in the way of making mathematical knowledge indubitable.\u201d\u00a0 (113)<\/p>\n

What are the implications?\u00a0 Guillen mentions three:<\/p>\n