[FOM] Foundational Challenge

[Richard Kimberly Heck]

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
<https://aapt.scitation.org/doi/abs/10.1119/1.2967702?journalCode=ajp>

[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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