Dear Sol,

This message is not specifically about your rebuttal of Hilary’s claim, but about a more general issue which I hope that you can shed light on.

You write:

Q1. Just which mathematical entities are indispensable to current scientific theories?, and

Q2. Just what principles concerning those entities are need for the required mathematics?

My very general question is: What do we hope to gain by showing that something can be “captured by limited means” (in this case regarding what mathematics is needed for physical theory)? Does this tell us something new about what we have “captured”?

I am of course familiar with advantages of, for example, establishing that some computable function is in fact provably total in PA, as then one might extract useful and new information about the growth rate of such a function. In set theory is something analagous, which is if you can bring down the large cardinal strength enough, core model theory kicks in and you have a good chance of achieving a much better understanding. Or if one starts with a philosophical position, like predicativity, it is somehow gratifying to know that one can capture it precisely with formal means.

But frankly speaking, too often there is a connotation of “of, we don’t really need all of that bad set theory to do this”, or even more outdated: “what a relief, now we know that this is consistent because we captured it in a system conservative over PA!”. Surely in the 21st century we are not going to worry anymore about the consistency of ZFC.

Is the point that (as you say at the end of your message) that you think you have to invoke some kind of platonistic ontology if you are not using limited means, and for some reason this makes you feel uncomfortable (even though I presume you don’t have inconsistency worries)?

It is tempting to think that your result using your system W might tell us something new about physics. Does it? On the other hand you have not claimed that “physics is conservative over PA” exactly, but only that the math needed to do a certain amount of physics is conservative over PA.

Finally, how is it that you claim that “only a platonistic philosophy of mathematics provides justification” for impredicative 2nd order arithmetic? That just seems wrong, as there are plenty of non-platonists out there (I am one) who are quite happy with ZFC. But maybe I don’t understand how you are using the word “justification”.

Thanks in advance for your clarifications. And please understand, I am not suggesting that it is not valuable to “capture things by limited means”, I just want to have a better understanding of what you feel is gained by doing that.

All the best,

Sy