Commitments (Arithmetic)

A thought without a resolution; a question without an answer I propose in the following:

Consider the following statements:

i. The set of real numbers has a subset of entities which are prime

ii. There is always a higher prime number after any given prime number n

iii. Either there is a highest prime number, or there is not

iv. (It is necessarily the case that) If I entertain any candidate for a ‘highest prime number’; there will always be a higher prime

v. If there is either a highest prime number or there is not, and it is the case that there will always be a higher prime, then it is not the case that there is a highest prime number

vi. If the number line were finite, then perhaps there would be a highest prime number

These are all different kinds of statements:

i. Is a claim about a set of mathematical objects, an assertoric claim about the number line; but is it existential? Is it analytic? is it a priori? is it contingent or necessary?

ii. The expression ‘There is’ intuitively expresses an existential proposition, however, it is an existential proposition about a set (the set of prime numbers), which itself is part of a set (natural numbers); so, to say ‘there is’, in this context is necessarily elliptical upon the committment to the prime number and real number set; do any of these have existential commitment within itself?

iii. Is a disjunctive logical statement; the nature of a disjunction (PvQ) is such that P may hold as an assertoric claim, where Q may be complete nonsense; furthermore, do we have an exclusive (PvQ), where it is the case that (P&Q); or (P&¬(Q)) v ((¬P)&Q)? furthermore, if we address this issue of the inclusive or exclusive disjunction, is this an ontic claim about the metalanguage? Further; what rules can we assert about the construction of the grammar, and what is the status of these rules themselves?

v. Is a modus ponens; to what do I commit to when I assert modus ponens? Do I need to commit to P to assert P–>Q? Furthermore; do I need to assert the truth claim; or tacitly suppose the rule of modus ponens in order to assert any material implicature

iv. Is a necessity claim, and thus, a modal claim; how do we construct necessity claims at all? Do we need token entities to instantiate natural kinds to demarcate the neccessary? Or do we need to posit an Ersatz plurality to ensure the plenitude of possibilities being genuinely assertoric?

vi. This is a counterfactual; is it absurd to posit a counterfactual about a finite number line? For example; if the last number in the line was 100, and there was nothing above it; is this inconcievable, further, is this possible, but inconceivable? or impossible because its inconceivable? Finally, if we were a Lewisian Modal Realist; could we entertain modal claims about the nature of mathematics itself? or modal claims about the rules of modality? (for instance, saying ‘it’s possible to have a world where we are ersatzist, and another world which is Lewisian in logical space’? This sounds ridiculous a claim prima facie.

Destre (and Michael)

One thought on “Commitments (Arithmetic)

  1. What about the status of sentences used to describe the behaviour of computers?

    What is the status of algorithms, for instance?

    All very difficult questions, Masters.


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