[Math] How one should treat M.Kline’s “Mathematics. The Loss of Certainty”

Question

Recently the article "Foundations of mathematics" in Russian Wikipedia attracted my attention by lots of strange (and often absurd) declarations, in particular, it is written there that David Hilbert (it is not clear, apparently, in some period of his life?) accepted the intuitionistic views.

When discussing this with the Wikipedia authors I understood that a large part of those oddities comes from the Morris Kline book "Mathematics: The Loss of Certainty". As an illustration, at page 250 (Oxford University Press, 1980) he writes that

In metamathematics, Hilbert proposed to use a special logic that was to be free of all objections. The logical principles would be so obviously true that everyone would accept them. Actually, they were very close to the intuitionist principles. Controversial reasoning–such as proof of existence by contradiction, transfinite induction, actually infinite sets, impredicative definitions, and the axiom of choice was not to be used.

Can anybody explain me what this can mean? Is it possible that Hilbert indeed agreed with intuitionists in some moment of his life? If yes, when was that, and when did he change his mind?

Or the explanaltion is that Kline simply does not understand what he describes (and therefore his book can't be treated as a reliable source)?

I would be grateful to people who could cast light on this because from what is written in the Wikipedia article it is seen that the declarations like those from the Kline book generated a series of further interpretations in other "popular texts", which led finally to absolutely absurd conlusions where, for example, Hilbert is presented as a loser, mathematics as a part of science that "abandoned claims for significance of its results", etc.

I can't read this, but I am not a specialist in history of mathematics, and it's difficult for me to understand what can lie behind all this. On the other hand the Wikipedia rules are contradictory, they give a possibility to the people who reached some power in its feudal stairs to abuse this power. So I need help.

EDIT. From the discussion in comments it became clear that the following detail could resolve the main part of my doubts:

Is it true that Hilbert agreed somewhere that the law of excluded middle (and the proofs by contradiction) must be rejected?

This sounds completely implausible.

Answer 1

Hilbert had a long career and, unsurprisingly, used different logics for different purposes. For his mathematical work, Hilbert is well known as a proponent of classical reasoning, including the law of the excluded middle and the axiom of choice.

For his consistency program, however, Hilbert referred to "finitary" methods. This program is described well in the SEP article. Hilbert did not formally define a logical system for finitism. He explained his motivation for not doing so in his speech "On the infinite" (1925), although his reasoning is still not completely clear to me:

In analyzing an existential statement whose content cannot be expressed by a finite disjunction, we encounter the infinite. Similarly, by negating a general statement, i.e., one which refers to arbitrary numerical symbols, we obtain a transfinite statement. For example, the statement that if a is a numerical symbol, then a + 1 = 1 + a is universally true, is from our finitary perspective incapable of negation. We will see this better if we consider that this statement cannot be interpreted as a conjunction of infinitely many numerical equations by means of `and' but only as a hypothetical judgment which asserts something for the case when a numerical symbol is given.

From our finitary viewpoint, therefore, we cannot argue that an equation like the one just given, where an arbitrary numerical symbol occurs, either holds for every symbol or is disproved by a counter example. Such an argument, being an application of the law of excluded middle, rests on the presupposition that the statement of the universal validity of such an equation is capable of negation.

At any rate, we note the following: if we remain within the domain of finitary statements, as indeed we must, we have as a rule very complicated logical laws. Their complexity becomes unmanageable when the expressions 'all' and 'there exists' are combined and when they occur in expressions nested within other expressions. In short, the logical laws which Aristotle taught and which men have used ever since they began to think do not hold. We could, of course, develop logical laws which do hold for the domain of finitary statements. But it would do us no good to develop such a logic, for we do not want to give up the use of the simple laws of Aristotelian logic. Furthermore, no one, though he speak with the tongues of angels, could keep people from negating general statements, or from forming partial judgments, or from using tertium non datur. What, then, are we to do?

...

It seems from this speech that Hilbert was at least partially concerned with the law of the excluded middle in the context of finitism as he understood it.

Modern formalizations of finitary reasoning typically do include the law of the excluded middle, although they can be weak in other ways. For example the theory of Primitive Recursive Arithmetic, often associated with finitism, is often presented as a theory with no quantifiers.

Separately, the work of Glivenko and Gödel in the 1930s showed that the law of the excluded middle on its own does not lead to contradiction. For example, Gödel proved that if first-order logic without excluded middle is consistent, then so is first-order logic with the law, and if Heyting Arithmetic without excluded middle is consistent then so is Peano Arithmetic, which consists of Heyting Arithmetic and the law of the excluded middle. In some settings, these results reduced the interest in the law of the excluded middle as a possible source of inconsistency. Of course, people may still use logics without LEM in order to ensure that proofs are more constructive or correspond more closely with algorithms.

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Regarding the book "Mathematics: the loss of certainty", I will simply quote the final paragraph of the review from the American Mathematical Monthly:

Finally, Professor Kline does not deal honestly with his readers. He is a learned man and knows perfectly well that many mathematical ideas created in abstracto have found significant application in the real world. He chooses to ignore this fact, acknowledged by even the most fanatic opponents of mathematics. He does this to support an untenable dogma. One is reminded of the story of the court jester to Louis XIV: the latter had written a poem and asked the jester his opinion. "Your majesty is capable of anything. Your majesty has set out to write doggerel and your majesty has succeeded". On balance, such, alas, must be said of this book.

This is unfortunate because other books, such as Kline's "Mathematical Thought from Ancient to Modern Times", do not have the same issues, and "The Loss of Certainty" can unfortunately cast a shadow on those as well.