Formal vs Contentual

Unlike the first three, there is no seminal paper that draws this contrast into sharp relief. Instead, the distinction becomes clear over the course of Hilbert’s papers. The formal/contentual distinction appears straightforward, but only in light of the ideal/concrete distinction. Once the contrast between ideal and concrete propositions is established, it is clear how to characterize Hilbert’s conception of formal versus contentual.

Formal reasoning is reasoning done solely through the manipulation of symbols according to a set of rigid rules. It is defined purely by its form, lacking any interpreted meaning behind the symbols: it is purely syntactic.

Contentual reasoning on the other hand, has content. There is an intended interpreted meaning. For Hilbert, a contentual proposition is one that attempts to communicate an idea to the reader and does so with an intended meaning. The mark ‘1’ is trivially formal, because it is a basic element of the theory, a concrete element devoid of meaning. Just as trivially, ‘2’ is contentual, since it has a meaning: it is intended to communicate to the reader the mark ‘11’. Similarly, ‘24’ is contentual because it means ‘111111111111111111111111’. Formal reasoning takes place when

and only when concrete objects are involved; contentual reasoning takes place in all other cases, but can be marked out positively by the presence of some identifiable meaning.

An example of the differentiation between contentual and formal reasoning will help illuminate this dichotomy—necessarily, the distinction mirrors the distinction between ideal and concrete. Hilbert [28] distinguishes between propositions like

1 + (1 + 1) = (1 + 1) + 1

and

1 + 2 = 2 + 1 and

a+b =b+a.

The first counts as a concrete, formal proposition. Its epistemic value derives from the fact that it is finite, surveyable, and epistemically secure. The second is also concrete, but strictly speaking, fails to be formal because the ‘2’ means something. It is a method of conveying a formal equation in a shorter, easier to understand manner. Its epistemic value trades on the fact that it communicates a finite, formal proposition, but is not a finite, formal proposition itself. Finally, the third proposition is composed of ‘a’ and ‘b’, which are uninterpreted formal objects without finite content. It is an ideal, transfinite proposition. Its epistemic value derives not from immediate grasp of its content but from its derivability from the transfinite axiom (page 65). Formal reasoning must involve only propositions that resemble the first equation. Contentual reasoning takes place involving equations that resemble the second and third equations.

proof-sketch of properly formal, finitary proofs, but does not formally prove anything. For instance, a proof of the statement 1 +a=a+ 1 can be accomplished through the application of the mathematical induction axiom. We show that 1 + 1 = 1 + 1 is true. Then, we assume that 1 +n =n+ 1 holds and show that 1 + (n+ 1) = (n+ 1) + 1 holds. Via the ideal principle of induction, we will have then shown that for alla, the statement ‘1 +a =a+ 1’ is true. But for Hilbert, the proof just given is not a real proof in the sense that it is not wholly formal and finite. It is merely a useful way of communicating both that there is a finite, purely formal proof of that equation and giving an outline of how to proceed.16 _{It is short hand for an infinite collection of}

real proofs, for example, the proof that shows that ‘1 + 57 = 57 + 1’ is true. Hilbert draws the distinction thus:

If we generalize this conception [of treating ideal propositions as com- municating finite propositions], mathematics becomes an inventory of formulas—first, formulas to which contentual communications of finitary propositions [hence in the main, numerical equations and inequalities] cor- respond and, second, further formulas that mean nothing in themselves and are the ideal objects of our theory[28, p.380, emphasis in original]

The first set of formulas Hilbert mentions here is the set of contentual propositions composed of numerals and normal mathematical symbols that serve to communicate formal propositions made up of concrete objects. Their referents are the subject of the theory make the goal of Hilbert’s project—the elimination of epistemic doubt in the foundations of mathematics—possible by their secure epistemic status. The second set of objects are the ideal propositions that provide the mathematical machinery to derive arithmetic from Hilbert’s foundation.

How do these four distinctions help to clarify Hilbert’s project in the 1920s? They do so by providing us a precise vocabulary to describe the details of Hilbert’s project.

16_{I should mention here that this interpretation of what Hilbert is doing in terms of the status of}

proofs using ideal propositions is somewhat controversial. Detlefsen disagrees, but I will argue later that Detlefsen’s reconstruction fails to avoid a serious Poincar´ean objection, so the question of its faithfulness to Hilbert in this regard becomes moot.

Hilbert must provide an axiomatic foundation and then prove two things about it: 1) that it actually is a foundation for mathematics, and 2) that the foundation is itself without doubt.17 _{The foundation that Hilbert provides is a finite, formal axiom}

system about concrete, purely syntactic marks. Because it deals only with these finite marks, Hilbert holds that it is beyond epistemic doubt.18 _{He proves that it is}

a foundation by reconstructing arithmetic via the introduction of ideal axioms to the theory, thereby allowing a contentual derivation of all mathematical results. However, the introduction of these transfinite ideal elements means that the theory has drifted from the epistemic safety of finitism into the danger of the transfinite. In order to prove that these axioms have not led the project astray, Hilbert creates a proof schema in the purely finite metatheory using solely finitary that the axioms can lead to no contradiction. The proof proceeds by demonstrating that any purported proof of a contradiction is not in obedience with the ideal axioms. Because any such purported proof will be finite, this proof sketch is a contentual demonstration of a set of finite proofs. Because of this finitism, Hilbert thinks the epistemic status of the theory is secure.