I agree with this point of view! My concern is with people who do not share this relatively benign view of infinity, and regard theorems proven using strong systems like ZF with suspicion.
The reason it matters what counts as “logic” is that such skeptics, to the extent that they can discuss matters of proof and soundness at all, must have a core of formally acceptable reasoning whose derived propositions they grant validity to, and therefore the parts of mathematics that can be interpreted in such a system overcome this skepticism.
Sep 9, 2023·edited Sep 9, 2023Liked by Joel David Hamkins
What is interesting is not “whether” mathematics can be reduced to logic, but “how much” of mathematics can be reduced to logic. There are several ways to get as far as PA that could be deemed “logical”, but doesn’t going further than that entail ontological commitments to infinite objects or collections of some kind?
Is there a plausible way to present second-order logic, or some other system that is arguably “logical”, where the existence of infinite entities follows naturally?
To my way of thinking, the main project of logicism can descend ultimately into unimportant (and uninteresting) squabbles about what counts as logic. What does it matter whether we label second-order arithmetic or set theory as logic or as something else? It seems eminently reasonable to me, for example, to count logicism as essentially fulfilled, almost completely, by the recognition that ZFC set theory serves very well as a foundation of mathematics. Essentially every mathematical idea can be interpreted in set theory, which seems capable of implementing essentially arbitrary abstract mathematical structure. And the philosophy of the cumulative hierarchy in set theory seems essentially logical in nature, logical at its core. In my view, the central axioms of set theory, including not only separation and power set, pairing, union etc., but also specifically including the axiom of choice and the axiom of infinity, are essentially "logical" in nature. (I recognize that some prominent people disagree with me about this.) The matter of choice is irrelevant in any case as ZFC is interpretable in ZF, and so we can get a foundation that way. Most of what Frege was trying to found in logic was finite mathematics and specifically natural number arithmetic, and for this, we can found it without the axiom of infinity. However, the existence of infinite sets seems fundamentally logical in nature to me, as much as the other kinds of logical argument Frege was undertaking. So all things considered, my view is that logicism is a success on the basis of the success of ZFC set theory as a foundation of mathematics.
I agree with this point of view! My concern is with people who do not share this relatively benign view of infinity, and regard theorems proven using strong systems like ZF with suspicion.
The reason it matters what counts as “logic” is that such skeptics, to the extent that they can discuss matters of proof and soundness at all, must have a core of formally acceptable reasoning whose derived propositions they grant validity to, and therefore the parts of mathematics that can be interpreted in such a system overcome this skepticism.
What is interesting is not “whether” mathematics can be reduced to logic, but “how much” of mathematics can be reduced to logic. There are several ways to get as far as PA that could be deemed “logical”, but doesn’t going further than that entail ontological commitments to infinite objects or collections of some kind?
Is there a plausible way to present second-order logic, or some other system that is arguably “logical”, where the existence of infinite entities follows naturally?
To my way of thinking, the main project of logicism can descend ultimately into unimportant (and uninteresting) squabbles about what counts as logic. What does it matter whether we label second-order arithmetic or set theory as logic or as something else? It seems eminently reasonable to me, for example, to count logicism as essentially fulfilled, almost completely, by the recognition that ZFC set theory serves very well as a foundation of mathematics. Essentially every mathematical idea can be interpreted in set theory, which seems capable of implementing essentially arbitrary abstract mathematical structure. And the philosophy of the cumulative hierarchy in set theory seems essentially logical in nature, logical at its core. In my view, the central axioms of set theory, including not only separation and power set, pairing, union etc., but also specifically including the axiom of choice and the axiom of infinity, are essentially "logical" in nature. (I recognize that some prominent people disagree with me about this.) The matter of choice is irrelevant in any case as ZFC is interpretable in ZF, and so we can get a foundation that way. Most of what Frege was trying to found in logic was finite mathematics and specifically natural number arithmetic, and for this, we can found it without the axiom of infinity. However, the existence of infinite sets seems fundamentally logical in nature to me, as much as the other kinds of logical argument Frege was undertaking. So all things considered, my view is that logicism is a success on the basis of the success of ZFC set theory as a foundation of mathematics.