## Tag Archives: Peano

### On Gödel’s Incompleteness Theorem

This is an appreciation of Gödel’s Incompleteness Theorem of 1931. I am provoked by a depreciation of the theorem.

I shall review the mathematics of the theorem, first in outline, later in more detail. The mathematics is difficult. I have trouble reproducing it at will and even just confirming what I have already written about it below (for I am adding these words a year after the original publication of this essay).

The difficulty of Gödel’s mathematics is part of the point of this essay. A person who thinks Gödel’s Theorem is unsurprising is probably a person who does not understand it.

In the “Gödel for Dummies” version of the Theorem, there are mathematical sentences that are both true and unprovable. This requires two points of clarification.

### Boolean Arithmetic

Mathematics can be highly abstract, even when it remains applicable to daily life. I want to show this with the mathematics behind logic puzzles, such as how to derive a conclusion using all of the following premisses:

1. Babies are illogical.
2. Nobody is despised who can manage a crocodile.
3. Illogical persons are despised.

The example, from Terence Tao’s blog, is attributed to Lewis Carroll. By the first and third premisses, babies are despised; by the second premiss then, babies cannot manage crocodiles.

George Boole, The Laws of Thought (1854), Open Court, 1940

### NL XIV: “Reason”

Index to this series

Summary added January 29, 2019, revised May 8, 2019. Practical reason is the support of one intention by another; theoretical, one proposition by another. Reasoning is thus always “motivated reasoning”: we engage in it to relieve the distress of uncertainty. Reason is primarily practical, only secondarily theoretical; and the reason for saying this is the persistence of anthropomorphism in theoretical reasoning: by the Law of Primitive Survivals in Chapter IX, we tend to think even of inanimate objects as forming intentions the way we do.

Reasons for adding this summary of Chapter XIV of Collingwood’s New Leviathan include

• the tortuousness of the following post on the chapter,
• the provocation of a Guardian column by Oliver Burkeman on motivated reasoning.

Says Burkeman, whose “problem” is apparently motivated reasoning itself,

One of the sneakier forms of the problem, highlighted in a recent essay by the American ethicist Jennifer Zamzow, is “solution aversion”: people judge the seriousness of a social problem, it’s been found, partly based on how appetising or displeasing they find the proposed solution. Obviously, that’s illogical …

On the contrary, how we reason cannot be “illogical,” any more than how we speak can be “ungrammatical.” Logic is an account, or an analysis, of how we do actually reason; grammar, of how we speak. Of course we may make errors, by our own standards.

Rogier van der Weyden (Netherlandish, 1399/1400–1464),
Portrait of a Lady, c. 1460, oil on panel
National Gallery of Art, Washington; Andrew W. Mellon Collection

### Context

There was a rumor that Collingwood had become a communist. According to David Boucher, editor of the revised (1992) edition of The New Leviathan, the rumor was one of the “many reasons why [that book] failed to attract the acclaim which had been afforded Collingwood’s other major works.” Continue reading