[FOM] Foundational Challenge

[Joe Shipman]

Riki, my analogy with physics is intended to highlight the professional concerns of researchers. Working physicists have no use for alternative interpretations of quantum mechanics because, so far, they have provided neither new predictions nor easier ways to get to known results.

I claim that alternative foundations for mathematics ought to meet a similar test—help working mathematicians find new results or reach old ones more easily.

Examples, please.

— JS

Sent from my iPhone

> On Jan 11, 2020, at 1:20 AM, Richard Kimberly Heck <richard_heck at brown.edu> wrote:
> 
> 
> On 1/9/20 2:44 PM, Michael Lee Finney wrote:
>>> Therefore I issue the following challenge to proponents of any kind of
>>> “alternative foundations” to “ZFC plus large cardinals”.
>>> I CHALLENGE YOU TO IDENTIFY:
>>> An EXAMPLE of mathematics done in any “alternative foundation” to set theory, where EITHER
>>> a) it is not obvious that any result derived that is statable in set theory
>>> will be provable in ZFC plus a large cardinal of some kind
>>> — JS
>> That is not the purpose of alternative foundations. 
> I do tend to agree with this, though no doubt there are others who would disagree. Perhaps it is worth pursuing the analogy with disputes over the interpretation of quantum mechanics a bit further. The issue there is not usually empirical. Everyone knows how to use the formalism of quantum mechanics to make predictions, etc. The problem---the `scandal' of quantum mechanics [1]---is that no one seems to know what the formalism *means*, that is, what it is saying about the fundamental nature of physical reality. 
> 
> There are a number of responses to this problem, one of which is to insist that it is a pseudo-problem, that scientific theories do not mean to tell us about the nature of reality, but only to make predictions about possible observations. Such views are known as varieties of `instrumentalism', and Hilbert's views are sometimes said to be an example of this variety, in mathematics [2]. But such views are obviously revisionist: Certainly most physicists think that quantum mechanics tells us something about the nature of physical reality that explains why the observable data are as they are. Instrumentalism utterly ignores the explanatory ambitions of scientific theories.
> 
> Is there an analogy in the case of mathematical foundations? I have not thought enough about mathematical explanation (let alone about its implications for foundational questions) to have a firm view about this matter. But it certainly does seem as if it might be understood in a similar way. Foundations might well be understood, it seems to me, as asking, very roughly: Why is the mathematical universe organized in such a way as to make our mathematical theories (predominantly) true of it? Possible answers are legion (including ones that reject the question).
> 
> Riki
> 
> [1] https://aapt.scitation.org/doi/abs/10.1119/1.2967702
> 
> [2] Harvey's work on the arithmetical consequences of large cardinal axioms might be thought of as an attempt to meet such instrumentalism on its own ground.
> 
> -- 
> ----------------------------
> Richard Kimberly (Riki) Heck
> Professor of Philosophy
> Brown University
> 
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