Platonist Assertory Mathematics. Thought of the Day 88.0


Traditional Platonism, according to which our mathematical theories are bodies of truths about a realm of mathematical objects, assumes that only some amongst consistent theory candidates succeed in correctly describing the mathematical realm. For platonists, while mathematicians may contemplate alternative consistent extensions of the axioms for ZF (Zermelo–Fraenkel) set theory, for example, at most one such extension can correctly describe how things really are with the universe of sets. Thus, according to Platonists such as Kurt Gödel, intuition together with quasi-empirical methods (such as the justification of axioms by appeal to their intuitively acceptable consequences) can guide us in discovering which amongst alternative axiom candidates for set theory has things right about set theoretic reality. Alternatively, according to empiricists such as Quine, who hold that our belief in the truth of mathematical theories is justified by their role in empirical science, empirical evidence can choose between alternative consistent set theories. In Quine’s view, we are justified in believing the truth of the minimal amount of set theory required by our most attractive scientific account of the world.

Despite their differences at the level of detail, both of these versions of Platonism share the assumption that mere consistency is not enough for a mathematical theory: For such a theory to be true, it must correctly describe a realm of objects, where the existence of these objects is not guaranteed by consistency alone. Such a view of mathematical theories requires that we must have some grasp of the intended interpretation of an axiomatic theory that is independent of our axiomatization – otherwise inquiry into whether our axioms “get things right” about this intended interpretation would be futile. Hence, it is natural to see these Platonist views of mathematics as following Frege in holding that axioms

. . . must not contain a word or sign whose sense and meaning, or whose contribution to the expression of a thought, was not already completely laid down, so that there is no doubt about the sense of the proposition and the thought it expresses. The only question can be whether this thought is true and what its truth rests on. (Frege to Hilbert Gottlob Frege The Philosophical and Mathematical Correspondence)

On such an account, our mathematical axioms express genuine assertions (thoughts), which may or may not succeed in asserting truths about their subject matter. These Platonist views are “assertory” views of mathematics. Assertory views of mathematics make room for a gap between our mathematical theories and their intended subject matter, and the possibility of such a gap leads to at least two difficulties for traditional Platonism. These difficulties are articulated by Paul Benacerraf (here and here) in his aforementioned papers. The first difficulty comes from the realization that our mathematical theories, even when axioms are supplemented with less formal characterizations of their subject matter, may be insufficient to choose between alternative interpretations. For example, assertory views hold that the Peano axioms for arithmetic aim to assert truths about the natural numbers. But there are many candidate interpretations of these axioms, and nothing in the axioms, or in our wider mathematical practices, seems to suffice to pin down one interpretation over any other as the correct one. The view of mathematical theories as assertions about a specific realm of objects seems to force there to be facts about the correct interpretation of our theories even if, so far as our mathematical practice goes (for example, in the case of arithmetic), any ω-sequence would do.

Benacerraf’s second worry is perhaps even more pressing for assertory views. The possibility of a gap between our mathematical theories and their intended subject matter raises the question, “How do we know that our mathematical theories have things right about their subject matter?”. To answer this, we need to consider the nature of the purported objects about which our theories are supposed to assert truths. It seems that our best characterization of mathematical objects is negative: to account for the extent of our mathematical theories, and the timelessness of mathematical truths, it seems reasonable to suppose that mathematical objects are non-physical, non- spatiotemporal (and, it is sometimes added, mind- and language-independent) objects – in short, mathematical objects are abstract. But this negative characterization makes it difficult to say anything positive about how we could know anything about how things are with these objects. Assertory, Platonist views of mathematics are thus challenged to explain just how we are meant to evaluate our mathematical assertions – just how do the kinds of evidence these Platonists present in support of their theories succeed in ensuring that these theories track the truth?


Pluralist Mathematics, Minimalist Philosophy: Hans Reichenbach. Drunken Risibility.


Hans Reichenbach relativized the notion of the constitutive a priori. The key observation concerns the fundamental difference between definitions in pure geometry and definitions in physical geometry. In pure geometry there are two kinds of definition: first, there are the familiar explicit definitions; second, there are implicit definitions, that is the kind of definition whereby such fundamental terms as ‘point’, ‘line’, and ‘surface’ are to derive their meaning from the fundamental axioms governing them. But in physical geometry a new kind of definition emerges – that of a physical (or coordinative) definition:

The physical definition takes the meaning of the concept for granted and coordinates to it a physical thing; it is a coordinative definition. Physical definitions, therefore, consist in the coordination of a mathematical definition to a “piece of reality”; one might call them real definitions. (Reichenbach, 8)

Now there are two important points about physical definitions. First, some such correlation between a piece of mathematics and “a piece of physical reality” is necessary if one is to articulate the laws of physics (e.g. consider “force-free moving bodies travel in straight lines”). Second, given a piece of pure mathematics there is a great deal of freedom in choosing the coordinative definitions linking it to “a piece of physical reality”, since… coordinative definitions are arbitrary, and “truth” and “falsehood” are not applicable to them. So we have here a conception of the a priori which (by the first point) is constitutive (of the empirical significance of the laws of physics) and (by the second point) is relative. Moreover, on Reichenbach’s view, in choosing between two empirically equivalent theories that involve different coordinative definitions, there is no issue of “truth” – there is only the issue of simplicity. In his discussion of Einstein’s particular definition of simultaneity, after noting its simplicity, Reichenbach writes: “This simplicity has nothing to do with the truth of the theory. The truth of the axioms decides the empirical truth, and every theory compatible with them which does not add new empirical assumptions is equally true.” (p 11)

Now, Reichenbach went beyond this and he held a more radical thesis – in addition to advocating pluralism with respect to physical geometry (something made possible by the free element in coordinative definitions), he advocated pluralism with respect to pure mathematics (such as arithmetic and set theory). According to Reichenbach, this view is made possible by the axiomatic conception of Hilbert, wherein axioms are treated as “implicit definitions” of the fundamental terms:

The problem of the axioms of mathematics was solved by the discovery that they are definitions, that is, arbitrary stipulations which are neither true nor false, and that only the logical properties of a system – its consistency, independence, uniqueness, and completeness – can be subjects of critical investigation. (p 3)

It needs to be stressed here that Reichenbach is extending the Hilbertian thesis concerning implicit definitions since although Hilbert held this thesis with regard to formal geometry he did not hold it with regard to arithmetic.

On this view there is a plurality of consistent formal systems and the notions of “truth” and “falsehood” do not apply to these systems; the only issue in choosing one system over another is one of convenience for the purpose at hand and this is brought out by investigating their metamathematical properties, something that falls within the provenance of “critical investigation”, where there is a question of truth and falsehood. This radical form of pluralism came to be challenged by Gödel’s discovery of the incompleteness theorems. To begin with, through the arithmetization of syntax, the metamathematical notions that Reichenbach takes to fall within the provenance of “critical investigation” were themselves seen to be a part of arithmetic. Thus, one cannot, on pain of inconsistency, say that there is a question of truth and falsehood with regard to the former but not the latter. More importantly, the incompleteness theorems buttressed the view that truth outstrips consistency. This is most clearly seen using Rosser’s strengthening of the first incompleteness theorem as follows: Let T be an axiom system of arithmetic that (a) falls within the provenance of “critical investigation” and (b) is sufficiently strong to prove the incompleteness theorem. A natural choice for such an axiom system is Primitive Recursive Arithmetic (PRA) but much weaker systems suffice, for example, IΔ0 + exp. Either of these systems can be taken as T. Assuming that T is consistent (something which falls within the provenance of “critical investigation”), by Rosser’s strengthening of the first incompleteness theorem, there is a Π01-sentence φ such that (provably within T + Con(T )) both T + φ and T + ¬φ are consistent. However, not both systems are equally legitimate. For it is easily seen that if a Π01-sentence φ is independent from such a theory, then it must be true. The point being that T is ∑10-complete (provably so in T). So, although T + ¬φ is consistent, it proves a false arithmetical statement.

Mathematical Reductionism: As Case Via C. S. Peirce’s Hypothetical Realism.


During the 20th century, the following epistemology of mathematics was predominant: a sufficient condition for the possibility of the cognition of objects is that these objects can be reduced to set theory. The conditions for the possibility of the cognition of the objects of set theory (the sets), in turn, can be given in various manners; in any event, the objects reduced to sets do not need an additional epistemological discussion – they “are” sets. Hence, such an epistemology relies ultimately on ontology. Frege conceived the axioms as descriptions of how we actually manipulate extensions of concepts in our thinking (and in this sense as inevitable and intuitive “laws of thought”). Hilbert admitted the use of intuition exclusively in metamathematics where the consistency proof is to be done (by which the appropriateness of the axioms would be established); Bourbaki takes the axioms as mere hypotheses. Hence, Bourbaki’s concept of justification is the weakest of the three: “it works as long as we encounter no contradiction”; nevertheless, it is still epistemology, because from this hypothetical-deductive point of view, one insists that at least a proof of relative consistency (i.e., a proof that the hypotheses are consistent with the frequently tested and approved framework of set theory) should be available.

Doing mathematics, one tries to give proofs for propositions, i.e., to deduce the propositions logically from other propositions (premisses). Now, in the reductionist perspective, a proof of a mathematical proposition yields an insight into the truth of the proposition, if the premisses are already established (if one has already an insight into their truth); this can be done by giving in turn proofs for them (in which new premisses will occur which ask again for an insight into their truth), or by agreeing to put them at the beginning (to consider them as axioms or postulates). The philosopher tries to understand how the decision about what propositions to take as axioms is arrived at, because he or she is dissatisfied with the reductionist claim that it is on these axioms that the insight into the truth of the deduced propositions rests. Actually, this epistemology might contain a short-coming since Poincaré (and Wittgenstein) stressed that to have a proof of a proposition is by no means the same as to have an insight into its truth.

Attempts to disclose the ontology of mathematical objects reveal the following tendency in epistemology of mathematics: Mathematics is seen as suffering from a lack of ontological “determinateness”, namely that this science (contrarily to many others) does not concern material data such that the concept of material truth is not available (especially in the case of the infinite). This tendency is embarrassing since on the other hand mathematical cognition is very often presented as cognition of the “greatest possible certainty” just because it seems not to be bound to material evidence, let alone experimental check.

The technical apparatus developed by the reductionist and set-theoretical approach nowadays serves other purposes, partly for the reason that tacit beliefs about sets were challenged; the explanations of the science which it provides are considered as irrelevant by the practitioners of this science. There is doubt that the above mentioned sufficient condition is also necessary; it is not even accepted throughout as a sufficient one. But what happens if some objects, as in the case of category theory, do not fulfill the condition? It seems that the reductionist approach, so to say, has been undocked from the historical development of the discipline in several respects; an alternative is required.

Anterior to Peirce, epistemology was dominated by the idea of a grasp of objects; since Descartes, intuition was considered throughout as a particular, innate capacity of cognition (even if idealists thought that it concerns the general, and empiricists that it concerns the particular). The task of this particular capacity was the foundation of epistemology; already from Aristotle’s first premisses of syllogism, what was aimed at was to go back to something first. In this traditional approach, it is by the ontology of the objects that one hopes to answer the fundamental question concerning the conditions for the possibility of the cognition of these objects. One hopes that there are simple “basic objects” to which the more complex objects can be reduced and whose cognition is possible by common sense – be this an innate or otherwise distinguished capacity of cognition common to all human beings. Here, epistemology is “wrapped up” in (or rests on) ontology; to do epistemology one has to do ontology first.

Peirce shares Kant’s opinion according to which the object depends on the subject; however, he does not agree that reason is the crucial means of cognition to be criticised. In his paper “Questions concerning certain faculties claimed for man”, he points out the basic assumption of pragmatist philosophy: every cognition is semiotically mediated. He says that there is no immediate cognition (a cognition which “refers immediately to its object”), but that every cognition “has been determined by a previous cognition” of the same object. Correspondingly, Peirce replaces critique of reason by critique of signs. He thinks that Kant’s distinction between the world of things per se (Dinge an sich) and the world of apparition (Erscheinungswelt) is not fruitful; he rather distinguishes the world of the subject and the world of the object, connected by signs; his position consequently is a “hypothetical realism” in which all cognitions are only valid with reservations. This position does not negate (nor assert) that the object per se (with the semiotical mediation stripped off) exists, since such assertions of “pure” existence are seen as necessarily hypothetical (that means, not withstanding philosophical criticism).

By his basic assumption, Peirce was led to reveal a problem concerning the subject matter of epistemology, since this assumption means in particular that there is no intuitive cognition in the classical sense (which is synonymous to “immediate”). Hence, one could no longer consider cognitions as objects; there is no intuitive cognition of an intuitive cognition. Intuition can be no more than a relation. “All the cognitive faculties we know of are relative, and consequently their products are relations”. According to this new point of view, intuition cannot any longer serve to found epistemology, in departure from the former reductionist attitude. A central argument of Peirce against reductionism or, as he puts it,

the reply to the argument that there must be a first is as follows: In retracing our way from our conclusions to premisses, or from determined cognitions to those which determine them, we finally reach, in all cases, a point beyond which the consciousness in the determined cognition is more lively than in the cognition which determines it.

Peirce gives some examples derived from physiological observations about perception, like the fact that the third dimension of space is inferred, and the blind spot of the retina. In this situation, the process of reduction loses its legitimacy since it no longer fulfills the function of cognition justification. At such a place, something happens which I would like to call an “exchange of levels”: the process of reduction is interrupted in that the things exchange the roles performed in the determination of a cognition: what was originally considered as determining is now determined by what was originally considered as asking for determination.

The idea that contents of cognition are necessarily provisional has an effect on the very concept of conditions for the possibility of cognitions. It seems that one can infer from Peirce’s words that what vouches for a cognition is not necessarily the cognition which determines it but the livelyness of our consciousness in the cognition. Here, “to vouch for a cognition” means no longer what it meant before (which was much the same as “to determine a cognition”), but it still means that the cognition is (provisionally) reliable. This conception of the livelyness of our consciousness roughly might be seen as a substitute for the capacity of intuition in Peirce’s epistemology – but only roughly, since it has a different coverage.

Rants of the Undead God: Instrumentalism. Thought of the Day 68.1


Hilbert’s program has often been interpreted as an instrumentalist account of mathematics. This reading relies on the distinction Hilbert makes between the finitary part of mathematics and the non-finitary rest which is in need of grounding (via finitary meta-mathematics). The finitary part Hilbert calls “contentual,” i.e., its propositions and proofs have content. The infinitary part, on the other hand, is “not meaningful from a finitary point of view.” This distinction corresponds to a distinction between formulas of the axiomatic systems of mathematics for which consistency proofs are being sought. Some of the formulas correspond to contentual, finitary propositions: they are the “real” formulas. The rest are called “ideal.” They are added to the real part of our mathematical theories in order to preserve classical inferences such as the principle of the excluded middle for infinite totalities, i.e., the principle that either all numbers have a given property or there is a number which does not have it.

It is the extension of the real part of the theory by the ideal, infinitary part that is in need of justification by a consistency proof – for there is a condition, a single but absolutely necessary one, to which the use of the method of ideal elements is subject, and that is the proof of consistency; for, extension by the addition of ideals is legitimate only if no contradiction is thereby brought about in the old, narrower domain, that is, if the relations that result for the old objects whenever the ideal objects are eliminated are valid in the old domain. Weyl described Hilbert’s project as replacing meaningful mathematics by a meaningless game of formulas. He noted that Hilbert wanted to “secure not truth, but the consistency of analysis” and suggested a criticism that echoes an earlier one by Frege – why should we take consistency of a formal system of mathematics as a reason to believe in the truth of the pre-formal mathematics it codifies? Is Hilbert’s meaningless inventory of formulas not just “the bloodless ghost of analysis? Weyl suggested that if mathematics is to remain a serious cultural concern, then some sense must be attached to Hilbert’s game of formulae. In theoretical physics we have before us the great example of a [kind of] knowledge of completely different character than the common or phenomenal knowledge that expresses purely what is given in intuition. While in this case every judgment has its own sense that is completely realizable within intuition, this is by no means the case for the statements of theoretical physics. Hilbert suggested that consistency is not the only virtue ideal mathematics has –  transfinite inference simplifies and abbreviates proofs, brevity and economy of thought are the raison d’être of existence proofs.

Hilbert’s treatment of philosophical questions is not meant as a kind of instrumentalist agnosticism about existence and truth and so forth. On the contrary, it is meant to provide a non-skeptical and positive solution to such problems, a solution couched in cognitively accessible terms. And, it appears, the same solution holds for both mathematical and physical theories. Once new concepts or “ideal elements” or new theoretical terms have been accepted, then they exist in the sense in which any theoretical entities exist. When Weyl eventually turned away from intuitionism, he emphasized the purpose of Hilbert’s proof theory, not to turn mathematics into a meaningless game of symbols, but to turn it into a theoretical science which codifies scientific (mathematical) practice. The reading of Hilbert as an instrumentalist goes hand in hand with a reading of the proof-theoretic program as a reductionist project. The instrumentalist reading interprets ideal mathematics as a meaningless formalism, which simplifies and “rounds out” mathematical reasoning. But a consistency proof of ideal mathematics by itself does not explain what ideal mathematics is an instrument for.

On this picture, classical mathematics is to be formalized in a system which includes formalizations of all the directly verifiable (by calculation) propositions of contentual finite number theory. The consistency proof should show that all real propositions which can be proved by ideal methods are true, i.e., can be directly verified by finite calculation. Actual proofs such as the ε-substitution procedure are of such a kind: they provide finitary procedures which eliminate transfinite elements from proofs of real statements. In particular, they turn putative ideal derivations of 0 = 1 into derivations in the real part of the theory; the impossibility of such a derivation establishes consistency of the theory. Indeed, Hilbert saw that something stronger is true: not only does a consistency proof establish truth of real formulas provable by ideal methods, but it yields finitary proofs of finitary general propositions if the corresponding free-variable formula is derivable by ideal methods.

Epistemological Constraints to Finitism. Thought of the Day 68.0


Hilbert’s substantial philosophical claims about the finitary standpoint are difficult to flesh out. For instance, Hilbert appeals to the role of Kantian intuition for our apprehension of finitary objects (they are given in the faculty of representation). Supposing one accepts this line of epistemic justification in principle, it is plausible that the simplest examples of finitary objects and propositions, and perhaps even simple cases of finitary operations such as concatenations of numerals can be given a satisfactory account.

Of crucial importance to both an understanding of finitism and of Hilbert’s proof theory is the question of what operations and what principles of proof should be allowed from the finitist standpoint. That a general answer is necessary is clear from the demands of Hilbert’s proof theory, i.e., it is not to be expected that given a formal system of mathematics (or even a single sequence of formulas) one can “see” that it is consistent (or that it cannot be a genuine derivation of an inconsistency) the way we can see, e.g., that || + ||| = ||| + ||. What is required for a consistency proof is an operation which, given a formal derivation, transforms such a derivation into one of a special form, plus proofs that the operation in fact succeeds in every case and that proofs of the special kind cannot be proofs of an inconsistency.

Hilbert said that intuitive thought “includes recursion and intuitive induction for finite existing totalities.” All of this in its application in the domain of numbers, can be formalized in a system known as primitive recursive arithmetic (PRA), which allows definitions of functions by primitive recursion and induction on quantifier-free formulas. However, Hilbert never claimed that only primitive recursive operations count as finitary. Although Hilbert and his collaborators used methods which go beyond the primitive recursive and accepted them as finitary, it is still unclear whether they (a) realized that these methods were not primitive recursive and (b) whether they would still have accepted them as finitary if they had. The conceptual issue is which operations should be considered as finitary. Since Hilbert was less than completely clear on what the finitary standpoint consists in, there is some leeway in setting up the constraints, epistemological and otherwise, an analysis of finitist operation and proof must fulfill. Hilbert characterized the objects of finitary number theory as “intuitively given,” as “surveyable in all their parts,” and said that their having basic properties must “exist intuitively” for us. This characterization of finitism as primarily to do with intuition and intuitive knowledge has been emphasized in that what can count as finitary on this understanding is not more than those arithmetical operations that can be defined from addition and multiplication using bounded recursion.

Rejecting the aspect of representability in intuition as the hallmark of the finitary; one could take finitary reasoning to be “a minimal kind of reasoning supposed by all non-trivial mathematical reasoning about numbers” and analyze finitary operations and methods of proof as those that are implicit in the very notion of number as the form of a finite sequence. This analysis of finitism is supported by Hilbert’s contention that finitary reasoning is a precondition for logical and mathematical, indeed, any scientific thinking.

|, ||, |||, ||||| . The Non-Metaphysics of Unprediction. Thought of the day 67.1


The cornerstone of Hilbert’s philosophy of mathematics was the so-called finitary standpoint. This methodological standpoint consists in a restriction of mathematical thought to objects which are “intuitively present as immediate experience prior to all thought,” and to those operations on and methods of reasoning about such objects which do not require the introduction of abstract concepts, in particular, require no appeal to completed infinite totalities.

Hilbert characterized the domain of finitary reasoning in a well-known paragraph:

[A]s a condition for the use of logical inferences and the performance of logical operations, something must already be given to our faculty of representation, certain extra-logical concrete objects that are intuitively present as immediate experience prior to all thought. If logical inference is to be reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that can neither be reduced to anything else nor requires reduction. This is the basic philosophical position that I consider requisite for mathematics and, in general, for all scientific thinking, understanding, and communication. [Hilbert in German + DJVU link here in English]

These objects are, for Hilbert, the signs. For the domain of contentual number theory, the signs in question are sequences of strokes (“numerals”) such as

|, ||, |||, ||||| .

The question of how exactly Hilbert understood the numerals is difficult to answer. What is clear in any case is that they are logically primitive, i.e., they are neither concepts (as Frege’s numbers are) nor sets. For Hilbert, the important issue is not primarily their metaphysical status (abstract versus concrete in the current sense of these terms), but that they do not enter into logical relations, e.g., they cannot be predicated of anything.

Sometimes Hilbert’s view is presented as if Hilbert claimed that the numbers are signs on paper. It is important to stress that this is a misrepresentation, that the numerals are not physical objects in the sense that truths of elementary number theory are dependent only on external physical facts or even physical possibilities. Hilbert made too much of the fact that for all we know, neither the infinitely small nor the infinitely large are actualized in physical space and time, yet he certainly held that the number of strokes in a numeral is at least potentially infinite. It is also essential to the conception that the numerals are sequences of one kind of sign, and that they are somehow dependent on being grasped as such a sequence, that they do not exist independently of our intuition of them. Only our seeing or using “||||” as a sequence of 4 strokes as opposed to a sequence of 2 symbols of the form “||” makes “||||” into the numeral that it is. This raises the question of individuation of stroke symbols. An alternative account would have numerals be mental constructions. According to Hilber, the numerals are given in our representation, but they are not merely subjective “mental cartoons”.

One version of this view would be to hold that the numerals are types of stroke-symbols as represented in intuition. At first glance, this seems to be a viable reading of Hilbert. It takes care of the difficulties that the reading of numerals-as-tokens (both physical and mental) faces, and it gives an account of how numerals can be dependent on their intuitive construction while at the same time not being created by thought.

Types are ordinarily considered to be abstract objects and not located in space or time. Taking the numerals as intuitive representations of sign types might commit us to taking these abstract objects as existing independently of their intuitive representation. That numerals are “space- and timeless” is a consequence that already thought could be drawn from Hilbert’s statements. The reason is that a view on which numerals are space- and timeless objects existing independently of us would be committed to them existing simultaneously as a completed totality, and this is exactly what Hilbert is objecting to.

It is by no means compatible, however, with Hilbert’s basic thoughts to introduce the numbers as ideal objects “with quite different determinations from those of sensible objects,” “which exist entirely independent of us.” By this we would go beyond the domain of the immediately certain. In particular, this would be evident in the fact that we would consequently have to assume the numbers as all existing simultaneously. But this would mean to assume at the outset that which Hilbert considers to be problematic.  Another open question in this regard is exactly what Hilbert meant by “concrete.” He very likely did not use it in the same sense as it is used today, i.e., as characteristic of spatio-temporal physical objects in contrast to “abstract” objects. However, sign types certainly are different from full-fledged abstracta like pure sets in that all their tokens are concrete.

Now what is the epistemological status of the finitary objects? In order to carry out the task of providing a secure foundation for infinitary mathematics, access to finitary objects must be immediate and certain. Hilbert’s philosophical background was broadly Kantian. Hilbert’s characterization of finitism often refers to Kantian intuition, and the objects of finitism as objects given intuitively. Indeed, in Kant’s epistemology, immediacy is a defining characteristic of intuitive knowledge. The question is, what kind of intuition is at play? Whereas the intuition involved in Hilbert’s early papers was a kind of perceptual intuition, in later writings it is identified as a form of pure intuition in the Kantian sense. Hilbert later sees the finite mode of thought as a separate source of a priori knowledge in addition to pure intuition (e.g., of space) and reason, claiming that he has “recognized and characterized the third source of knowledge that accompanies experience and logic.” Hilbert justifies finitary knowledge in broadly Kantian terms (without however going so far as to provide a transcendental deduction), characterizing finitary reasoning as the kind of reasoning that underlies all mathematical, and indeed, scientific, thinking, and without which such thought would be impossible.

The simplest finitary propositions are those about equality and inequality of numerals. The finite standpoint moreover allows operations on finitary objects. Here the most basic is that of concatenation. The concatenation of the numerals || and ||| is communicated as “2 + 3,” and the statement that || concatenated with ||| results in the same numeral as ||| concatenated with || by “2 + 3 = 3 + 2.” In actual proof-theoretic practice, as well as explicitly, these basic operations are generalized to operations defined by recursion, paradigmatically, primitive recursion, e.g., multiplication and exponentiation. Roughly, a primitive recursive definition of a numerical operation is one in which the function to be defined, f , is given by two equations

f(0, m) = g(m)

f(n′, m) = h(n, m, f(n, m)),

where g and h are functions already defined, and n′ is the successor numeral to n. For instance, if we accept the function g(m) = m (the constant function) and h(n, m, k) = m + k as finitary, then the equations above define a finitary function, in this case, multiplication f (n, m) = n × m. Similarly, finitary judgments may involve not just equality or inequality but also basic decidable properties, such as “is a prime.” This is finitarily acceptable as long as the characteristic function of such a property is itself finitary: For instance, the operation which transforms a numeral to | if it is prime and to || otherwise can be defined by primitive recursion and is hence finitary. Such finitary propositions may be combined by the usual logical operations of conjunction, disjunction, negation, but also bounded quantification. The problematic finitary propositions are those that express general facts about numerals such as that 1 + n = n + 1 for any given numeral n. It is problematic because, for Hilbert it is from the finitist point of view incapable of being negated. By this he means that the contradictory proposition that there is a numeral n for which 1 + n ≠ n + 1 is not finitarily meaningful. A finitary general proposition is not to be understood as an infinite conjunction but only as a hypothetical judgment that comes to assert something when a numeral is given. Even though they are problematic in this sense, general finitary statements are of particular importance to Hilbert’s proof theory, since the statement of consistency of a formal system T is of such a general form: for any given sequence p of formulas, p is not a derivation of a contradiction in T. Even though in general existential statements are not finitarily meaningful, they may be given finitary meaning if the witness is given by a finitary function. For instance, the finitary content of Euclid’s theorem that for every prime p there is a prime > p, is that given a specific prime p one can produce, by a finitary operation, another prime > p (viz., by testing all numbers between p and p! + 1.).

ε-calculus and Hilbert’s Contentual Number Theory: Proselytizing Intuitionism. Thought of the Day 67.0


Hilbert came to reject Russell’s logicist solution to the consistency problem for arithmetic, mainly for the reason that the axiom of reducibility cannot be accepted as a purely logical axiom. He concluded that the aim of reducing set theory, and with it the usual methods of analysis, to logic, has not been achieved today and maybe cannot be achieved at all. At the same time, Brouwer’s intuitionist mathematics gained currency. In particular, Hilbert’s former student Hermann Weyl converted to intuitionism.

According to Hilbert, there is a privileged part of mathematics, contentual elementary number theory, which relies only on a “purely intuitive basis of concrete signs.” Whereas the operating with abstract concepts was considered “inadequate and uncertain,” there is a realm of extra-logical discrete objects, which exist intuitively as immediate experience before all thought. If logical inference is to be certain, then these objects must be capable of being completely surveyed in all their parts, and their presentation, their difference, their succession (like the objects themselves) must exist for us immediately, intuitively, as something which cannot be reduced to something else.

The objects in questions are signs, both numerals and the signs that make up formulas a formal proofs. The domain of contentual number theory consists in the finitary numerals, i.e., sequences of strokes. These have no meaning, i.e., they do not stand for abstract objects, but they can be operated on (e.g., concatenated) and compared. Knowledge of their properties and relations is intuitive and unmediated by logical inference. Contentual number theory developed this way is secure, according to Hilbert: no contradictions can arise simply because there is no logical structure in the propositions of contentual number theory. The intuitive-contentual operations with signs form the basis of Hilbert’s meta-mathematics. Just as contentual number theory operates with sequences of strokes, so meta-mathematics operates with sequences of symbols (formulas, proofs). Formulas and proofs can be syntactically manipulated, and the properties and relationships of formulas and proofs are similarly based in a logic-free intuitive capacity which guarantees certainty of knowledge about formulas and proofs arrived at by such syntactic operations. Mathematics itself, however, operates with abstract concepts, e.g., quantifiers, sets, functions, and uses logical inference based on principles such as mathematical induction or the principle of the excluded middle. These “concept-formations” and modes of reasoning had been criticized by Brouwer and others on grounds that they presuppose infinite totalities as given, or that they involve impredicative definitions. Hilbert’s aim was to justify their use. To this end, he pointed out that they can be formalized in axiomatic systems (such as that of Principia or those developed by Hilbert himself), and mathematical propositions and proofs thus turn into formulas and derivations from axioms according to strictly circumscribed rules of derivation. Mathematics, to Hilbert, “becomes an inventory of provable formulas.” In this way the proofs of mathematics are subject to metamathematical, contentual investigation. The goal of Hilbert is then to give a contentual, meta-mathematical proof that there can be no derivation of a contradiction, i.e., no formal derivation of a formula A and of its negation ¬A.

Hilbert and Bernays developed the ε-calculus as their definitive formalism for axiom systems for arithmetic and analysis, and the so-called ε-substitution method as the preferred approach to giving consistency proofs. Briefly, the ε-calculus is a formalism that includes ε as a term-forming operator. If A(x) is a formula, then εxA(x) is a term, which intuitively stands for a witness for A(x). In a logical formalism containing the ε-operator, the quantifiers can be defined by: ∃x A(x) ≡ A(εxA(x)) and ∀x A(x) ≡ A(εx¬A(x)). The only additional axiom necessary is the so-called “transfinite axiom,” A(t) → A(εxA(x)). Based on this idea, Hilbert and his collaborators developed axiomatizations of number theory and analysis. Consistency proofs for these systems were then given using the ε-substitution method. The idea of this method is, roughly, that the ε-terms εxA(x) occurring in a formal proof are replaced by actual numerals, resulting in a quantifier-free proof. Suppose we had a (suitably normalized) derivation of 0 = 1 that contains only one ε-term εxA(x). Replace all occurrences of εxA(x) by 0. The instances of the transfinite axiom then are all of the form A(t) → A(0). Since no other ε-terms occur in the proof, A(t) and A(0) are basic numerical formulas without quantifiers and, we may assume, also without free variables. So they can be evaluated by finitary calculation. If all such instances turn out to be true numerical formulas, we are done. If not, this must be because A(t) is true for some t, and A(0) is false. Then replace εxA(x) instead by n, where n is the numerical value of the term t. The resulting proof is then seen to be a derivation of 0 = 1 from true, purely numerical formulas using only modus ponens, and this is impossible. Indeed, the procedure works with only slight modifications even in the presence of the induction axiom, which in the ε-calculus takes the form of a least number principle: A(t) → εxA(x) ≤ t, which intuitively requires εxA(x) to be the least witness for A(x).

Weyl and Automorphism of Nature. Drunken Risibility.


In classical geometry and physics, physical automorphisms could be based on the material operations used for defining the elementary equivalence concept of congruence (“equality and similitude”). But Weyl started even more generally, with Leibniz’ explanation of the similarity of two objects, two things are similar if they are indiscernible when each is considered by itself. Here, like at other places, Weyl endorsed this Leibnzian argument from the point of view of “modern physics”, while adding that for Leibniz this spoke in favour of the unsubstantiality and phenomenality of space and time. On the other hand, for “real substances” the Leibnizian monads, indiscernability implied identity. In this way Weyl indicated, prior to any more technical consideration, that similarity in the Leibnizian sense was the same as objective equality. He did not enter deeper into the metaphysical discussion but insisted that the issue “is of philosophical significance far beyond its purely geometric aspect”.

Weyl did not claim that this idea solves the epistemological problem of objectivity once and for all, but at least it offers an adequate mathematical instrument for the formulation of it. He illustrated the idea in a first step by explaining the automorphisms of Euclidean geometry as the structure preserving bijective mappings of the point set underlying a structure satisfying the axioms of “Hilbert’s classical book on the Foundations of Geometry”. He concluded that for Euclidean geometry these are the similarities, not the congruences as one might expect at a first glance. In the mathematical sense, we then “come to interpret objectivity as the invariance under the group of automorphisms”. But Weyl warned to identify mathematical objectivity with that of natural science, because once we deal with real space “neither the axioms nor the basic relations are given”. As the latter are extremely difficult to discern, Weyl proposed to turn the tables and to take the group Γ of automorphisms, rather than the ‘basic relations’ and the corresponding relata, as the epistemic starting point.

Hence we come much nearer to the actual state of affairs if we start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations. Once the group is known, we know what it means to say of a relation that it is objective, namely invariant with respect to Γ.

By such a well chosen constitutive stipulation it becomes clear what objective statements are, although this can be achieved only at the price that “…we start, as Dante starts in his Divina Comedia, in mezzo del camin”. A phrase characteristic for Weyl’s later view follows:

It is the common fate of man and his science that we do not begin at the beginning; we find ourselves somewhere on a road the origin and end of which are shrouded in fog.

Weyl’s juxtaposition of the mathematical and the physical concept of objectivity is worthwhile to reflect upon. The mathematical objectivity considered by him is relatively easy to obtain by combining the axiomatic characterization of a mathematical theory with the epistemic postulate of invariance under a group of automorphisms. Both are constituted in a series of acts characterized by Weyl as symbolic construction, which is free in several regards. For example, the group of automorphisms of Euclidean geometry may be expanded by “the mathematician” in rather wide ways (affine, projective, or even “any group of transformations”). In each case a specific realm of mathematical objectivity is constituted. With the example of the automorphism group Γ of (plane) Euclidean geometry in mind Weyl explained how, through the use of Cartesian coordinates, the automorphisms of Euclidean geometry can be represented by linear transformations “in terms of reproducible numerical symbols”.

For natural science the situation is quite different; here the freedom of the constitutive act is severely restricted. Weyl described the constraint for the choice of Γ at the outset in very general terms: The physicist will question Nature to reveal him her true group of automorphisms. Different to what a philosopher might expect, Weyl did not mention, the subtle influences induced by theoretical evaluations of empirical insights on the constitutive choice of the group of automorphisms for a physical theory. He even did not restrict the consideration to the range of a physical theory but aimed at Nature as a whole. Still basing on his his own views and radical changes in the fundamental views of theoretical physics, Weyl hoped for an insight into the true group of automorphisms of Nature without any further specifications.