Oversimplification: Newton & Leibnitz &c build the calculus and other methods that solve a vast number of practical problems. Weierstrass, Dedekind, Cantor &c build a foundation under it dependent on transfinite quantities.

Kronecker, Brouwer, etc, were appalled by this. Later, Bishop &c actually demonstrates approaches to founding these techniques on constructive methods.

In the long interval before Bishop, what did Intuitionists and/or Constructivists think about practical applications? Did they expect bridges to fall down? Or did they simply believe that mathematics had not yet built a meaningful foundation for the practical methods?

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    $\begingroup$ Et cetera non et cetra. In fact, in this case, et ceteri, as the poor guys are not objects. However, one usually abbreviates &c. $\endgroup$ – Pietro Majer Aug 7 '18 at 7:32
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    $\begingroup$ It should be remarked that by now, by work of Ye, we know that finitistic foundations suffice for nearly all of the advanced mathematics used in physical theories. I don't recall how constructive it all is offhand, but I think it's Bishop-adjacent, at least. $\endgroup$ – David Roberts Aug 7 '18 at 20:27

What did the logicians of the 20th century think? Perhaps this was best described by Michael Beeson in his book ("Foundations of constructive mathematics: metamathematical studies", 1985, Springer):

The thrust of Bishop's work was that both Hilbert and Brouwer had been wrong about an important point on which they had agreed. Namely both of them thought that if one took constructive mathematics seriously, it would be necessary to "give up" the most important parts of modern mathematics (such as, for example, measure theory or complex analysis). Bishop showed that this was simply false, and in addition that it is not necessary to introduce unusual assumptions that appear contradictory to the uninitiated. The perceived conflict between power and security was illusory! One only had to proceed with a certain grace, instead of with Hilbert's "boxer's fists".

So, apparently they were both too pessimistic. Of course, they were not so naive as to think that bridges would fall down if a switch to constructive foundations were made. They just expected the constructive foundations to be insufficient (in the same sense as, say, ancient Greeks' notion of number was insufficient for carrying out real analysis).

Let me also say that it is not a trivial matter to establish what sort of foundations were used by Newton, Leibniz, Weierstraß, Cauchy, etc., because they did not present their work in the way we usually do today. It is perhaps most honest to just admit that their notion of foundation was not of the same kind as ours. Nevertheless, at least all the calculational proofs (and that would be a large proportion of all proofs) would be naturally constructive already. Fundamentally non-constructive reasoning came later, with Hilbert's proof of Nullstellensatz and the non-constructive principles of set theory (such as the axiom of choice).

And one should not commit the logical error of thinking that just because pre-20th century mathematics could be retrofitted with classical first-order logic and Zermelo-Fraenkel set theory, it must be so retrofitted.

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    $\begingroup$ the constrictive foundations to be insufficient <-- intentional? $\endgroup$ – David Roberts Aug 7 '18 at 8:31
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    $\begingroup$ Andrej, you write: 'Namely both of them thought that if one took constructive mathematics seriously, it would be necessary to "give up" the most important parts of modern mathematics (such as, for example, measure theory or complex analysis).' This is not true, Brouwer did not think this at all. He worked on measure theory himself, and was proficient in real and complex analysis, enough to have constructivized convincing samples of these. $\endgroup$ – Frank Waaldijk Aug 7 '18 at 8:53
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    $\begingroup$ "And if one succeeds in the construction of linguistic buildings, sequences of sentences proceeding according to the logical laws, thereby departing from linguistic images which could accompany basic mathematical truths in actual mathematical buildings, and if it turns out that those linguistic buildings can never produce the linguistic form of a contradiction, then all the same they belong to mathematics only in their quality of a linguistic building, and have nothing to do with mathematics outside of that building, e.g. with ordinary arithmetic or geometry. [...] $\endgroup$ – Noam Zeilberger Aug 7 '18 at 9:30
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    $\begingroup$ "So the idea that by means of such linguistic buildings we can obtain any knowledge of mathematics apart from that which can be constructed directly on the basis of intuition, is mistaken. And more so is the idea that in this way we can lay the foundations of mathematics, in other words that we can ensure the reliability of the mathematical theorems." (Brouwer, 1975, pp. 132–133, original emphasis) $\endgroup$ – Noam Zeilberger Aug 7 '18 at 9:31
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    $\begingroup$ @AndrejBauer I am not a Brouwer scholar, but the way I read the passage is that he clearly places value in the ordinary objects of the mathematical world ("ordinary arithmetic or geometry"), and is merely skeptical about certain purported ways of studying them via specific formal systems. I see nothing to suggest that he would want to exclude these classical subjects from the scope of intuitionistic mathematics, and quite the contrary otherwise he wouldn't be so worried about "improper" foundations. $\endgroup$ – Noam Zeilberger Aug 7 '18 at 17:33

There is a more general question lurking in the background, which is what do critics of logical foundations generally think about applications?

Historically, intuitionism is not the only foundational controversy. Earlier, there were critics of the logical foundations of calculus (recall Berkeley's "ghosts of departed quantities"), and today there are unresolved mathematical difficulties in quantum field theory. The pattern is usually the same. The practitioners, based on intuition and experience, know what they have to do to make sure that "bridges don't fall down," while the critics point out that the practitioners have failed to articulate clearly what the ground rules are. In particular, the critics can sometimes construct calculations that appear to avoid all explicitly forbidden operations, yet yield the wrong answer.

To determine what a specific person (e.g., Brouwer) thought, one obviously needs to examine what that particular person said on the subject. But in general, there will be a range of opinions. Some, as you say, will believe that the scientific/engineering theories must be fundamentally correct even though we haven't hammered out all the logical details yet, while others of a more alarmist bent may worry that people are trusting the theories too much, to the point where a "bridge will fall down."

Obviously, in the real world, bridges do sometimes fall down, but there are many reasons for this, most of which have nothing to do with inadequate logical hygiene. I would be curious to know if there are any examples of an actual engineering disaster with the following features:

  1. The disaster can be traced to a calculation that the engineers mistakenly trusted (and not because there was an inadvertent error or bug).

  2. If one were to apply "more rigorous reasoning" then the calculation would have come out differently and the disaster would have been averted.

I can imagine that there might be modern examples where the results of some kind of numerical simulation are trusted, but where theoreticians can show that the numerical results do not accurately reflect the behavior of the equations. (If the equations themselves are an inadequate model of physical reality, then that is a different matter, which I'm not concerned with here.) But in general, I think such situations are rare, because theoretical predictions are usually tested experimentally before an actual engineering project that might endanger people's lives is carried out. If I am right about this then most fears of "bridges falling down" because of lack of logical rigor are overblown.

  • $\begingroup$ Tim, I think a concern about the lack of logical foundations as a cause for a "bridge to fall" is completely ridiculous. An interesting practical example where numerical simulations (not logical foundations) were not used properly, leading to a structural failure, is the Sleipner oil platform disaster, which is easily found with Google. $\endgroup$ – KConrad Aug 7 '18 at 21:38

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