I recently came across this paper I wrote for a Philosophy of Math class as an undergrad at UC Irvine. Although this heady topic has been addressed in greater depths by far greater minds- I thought I would share. In the very least it is a superficial introduction to a deeply important debate that is still actively discussed.

Frege, Hilbert, Intuition and the Truth of Axioms

At the turn of the nineteenth century, a great debate erupted in philosophy and mathematics concerning the proper application of geometry, intuition and the coherence of axioms. This eruption, embodied by the clash between two intellectual giants, ultimately spurred the development of logic toward Modal Theory, or the study of the relationships between formal languages and their interpretations. On the one hand, David Hilbert claimed that if the given axioms, along with all of their consequences, do not contradict one another then they must, in some sense, define mathematical truth. Importantly, his claim allowed for axioms to be true under certain interpretations, yet false under others. So terms, geometric or otherwise, could bear different meanings in different systems. Thus, his assertion that a concept can be fixed only by its relations to other concepts, and that “only the whole structure of axioms yields a complete definition” became the basis for modern stucturalism. Frege, on the other hand, argued that axioms express intuitive truths using concepts we already have. In other words, axioms are contentful propositions that are wholly different from definitions; they are factual statements about mathematical objects. Subsequently, he believed that our mathematical concepts must reduce to logical ones. This argument was based on the aforementioned assumption that we could have intuitive knowledge of our logical truths, and a pre-theoretic understanding of our primitive terms. Subsequently, Frege viewed Hilbert’s analysis as confused, misguided and lacking in clarity.

The purpose of this paper, however, is not to side with those that think Frege had historical blindness to Hilbert’s logical revolution- as I see valid points being made by both parties. Rather it will be to clarify their differences. For it is not hard to mount a defense of Frege’s epistemological claims about mathematics as understood within the confines of his first order logic. As such, I will argue that there is a distinct difference between semantic (or model-theoretic) and syntactic (or proof-theoretic) treatments of logical consequence. Frege’s view is most certainly semantic since it deals with meaning and not metatheory and simple formal structures. Hilbert’s, contrarily is defiantly syntactical.

As such, lets engage in a broader discussion of what Frege posited. It is often argued that since Frege relied on classical ideas of mathematics and geometry, he abruptly rejected any general notion that an axiom could be false. But that’s not quite precise. Yes he believed that no false proposition could be an axiom- but he also believed an axiom could be false if it were chosen poorly. His unassailable endorsement was that the axioms that mathematicians had actually chosen were correct and true. For this reason, Frege’s analysis produced a strong modality. Under his doctrine of sense and reference, a false axiom would automatically be excluded from the domain of contentful propositions. And consequently, since axioms are propositions- what holds for propositions holds for axioms. So, to consider an axiom as anything but true would be to consider it as having a different reference- and hence a different thought. This was the root of Frege’s issue with Hilbert. He states, “As if it were permissable to have different propositions with the same wording! Rubbish! A mere wording without a thought-content can never be proved.” (Letter, Frege to Hilbert, 1889) In other words, for Frege, if the axioms were uninterrupted they simply couldn’t be true because they didn’t say anything. He felt that any change made within an axiomatic system would not just yield a different interpretation of the language, but rather a wholly different language altogether. If axioms were left interpreted they could not bear truth since they did not express anything.

Continuing, Frege believed that axioms were logical laws because they encompassed “a thought whose truth is certain without being provable by a chain of logical inferences.” (Frege, On the Foundations of Geometry, Pg. 69) This simple notion that underscored the bulk of his dispute with Hilbert is present throughout their correspondence. Frege states, “ there can be talk about relations between concepts…only after these concepts have been given sharp limits, but not while they are being defined.” (Letter, Frege to Hilbert) Frege thought Hilbert was conflating the two very different concepts of definition and axiom.

Hilbert, contrarily, conceptualized the notion of truth in a structure, and that primitive terms could bear different meanings simply because a formal language is inherently uninterrupted. In other words, the axioms could be understood as meaningless inscriptions. He state, “a concept can be fixed logically only by its relations to other concepts.” (Letter, Hilbert to Frege, 1/6/1900) In other words, if axioms are satisfiable, they essentially characterize a structure, and are therefore true of it. Hilbert forwarded this claim through a discussion of non-Euclidian geometries, and in particular the provable negation of the axiom of parallels. The mere fact that there exist geometries in which postulates one through four are proven true while postulate five is proven false established his conjecture. He states, “The axiom of Parallels is independent of the other axioms. It can be seen that in the non-Euclidian geometry all axioms except axiom 5 are valid and since the existence of ordinary geometry has been proved…the existence of non-Euclidian follows now.” (Hilbert, Foundations of Geometry, Pg. 32-33)

Hilbert strengthened his case through the development of his vastly influential independency proofs. In Grundlangen der Geometrie he successfully established that his system of axiomatization was optimal since none of the axioms is ever a consequent of the remaining ones. Thus, the question of the consistency of his system can be presented formally as the query of whether there are proofs in the system of two contradictory results. Remember, it is a crucial assumption of Hilbert’s that a language does not come with fixed interpretations. This was a revolutionary point that could have not been more foreign to Frege. Frege states, “Here the axioms are made to carry a burden that belongs to definitions. To me this seems to obliterate the dividing line between definitions and axioms in a dubious manner, and beside the old meaning of the word “axiom”, which comes out in the proposition that the axioms express fundamental facts of intuition, there emerges another meaning, but one I can no longer grasp.” (Letter, Frege to Hilbert, 12/27/1899)

As such, a discussion on how Frege viewed independence is due. Frege derived it as thus: Axiom P is independent of of Axioms P1,…,Pn, if it can be assumed, without contradiction, that a replacement for P is false while P1,…Pn are true. So, a proposition is independent of a set of propositions if it isn’t a consequence of that set, and a proposition or a set of propositions, is consistent if no contradiction is a consequence of it. Applicably, independence in logic could be utilized symbolically by showing that a set consisting of several axioms plus the negation of another axiom would not turn up any contradiction. In other words, sigma is independent of phi if there is no sequence of logical steps that leads from phi to sigma. Importantly, this template required relations to be substituted from the same logical category (names to names, nouns to nouns, functions to functions of the corresponding arity) and have a fixed meaning and a fixed reference beforehand. Following, Frege posited that if we can think of it semantically, we can show a lack of contradiction. Yet, despite the development of his complex system designed merely to demonstrate that one claim is a consequent of others, Frege never offered up a proof of independence of consistency. Why? Because although he made a firm commitment to the aforementioned consistency claims, he did not believe that they needed to be demonstrated systematically. He argued instead that we know the axioms of geometry are true and that consistency follows from truth. He states, “What I call a proposition tout court or a real proposition is a group of signs that expresses a thought; however, whatever only has the grammatical form of a proposition I call a pseudo-proposition.” (Frege, On the Foundations of Geometry, Pg 69) Thus, clearly, from Frege’s point of view, Hilbert was trading in pseudo-propositions.

Hilbert, on the other hand, was able to illustrate his logical intentions syntactically through the development of geometric groups of axioms which were ultimately held independent of each other. Each of these groups were further defined by a concept such as Uniqueness, Betweeness, or Incidence. And, depending which group of statements were included among the axioms, different definitions of plane, point, or line would arise, which were, in turn, implicitly defined by the axioms held to be true. This was possible since Hilbert conceived of these axioms as being only partially interpreted. So, although the logical terms in these propositions would have fixed meanings, the non-logical terms would be wholly schematic and have no fixed semantical value. Following, a geometric statement might be considered either true of false depending upon the implicit definitions of the geometrical non-logical terms given. Thus, under his view, the non-logical terms are open to assignments of properties, relations, or sets.

Consider first the Uniqueness Axiom: xyz{(B xyz V B xyz) V B xyz)} which states that for any three points xyz on a line, exactly one, and at most one, of the points lies between the other two. So if L and M are distinct lines that are not parallel, then L and M have a unique point in common B. And if we suppose that L and M have two distinct points in common A and B, by the Line Uniqueness Axiom, L and M must be the same line which is a contradiction to the hypothesis. Ergo, A and B must be the same point.

Next let’s consider the group of Hilbert axioms that define Betweeness. If A*B*C*, then A, B, C are three distinct points all lying on the same line, and C*B*A. Given any two distinct points B and D, there exists A, C and E lying on line segment BD such that A*B*D*, and B*C*D* and line segment BC. If A, B and C are three distinct points lying on the same line, then one and only one of the points is between the other two. Following, since Hilbert took this notion of Betweeness represented by “the point B is between the points A and C”, to be an undefined term, he was able to further induce the axiom of Separation. If A, B and C are three distinct points lying on the same line L, then one and only one of the points is between the other two. And if A and B are on the same side of L and B and C are on the same side of L, then A and C are on the same side of L. If A and B are on opposite sides of L, and B and C are on opposite sides of L, then A and C are on the same side of L. So, Separation basically turned Betweeness into a four place relation (S xyzu) predicated on the notion that a pair of points could satisfy Betweeness when Uniqueness failed. Thus, for any of the points xyzu “there is a unique division of them into pairs, such that one pair separates the other.” (Hilbert)

When it came to intuition, Frege and Hilbert shared some common ground. This was due in part to the fact that they were both Kantians. For example, they both made an appeal to the use of a priori justification when it came to the selection of axioms. In Grundlangen der Geometrie Hilbert states that axioms express “facts of our intuition.” (Pg. 3) Additionally, they both agreed that the axiomatic method was superior to the genetic method of derivation since “the genetic method leaves something to be desired as far as logical certainty is concerned.” (Letter, Frege to Hilbert, 1/6/1900) So, by advocating that axiomization is the route to guaranteed truth, they both felt that traditional logic was severely inadequate. Hilbert remarked to Frege, “I believe that your view of the nature and purpose of symbolism in mathematics is exactly right. I agree especially that the symbolism must come later and in response to a need, from which it follows, of course, that whoever wants to create or develop a symbolism must first study those needs.” (Letter, Hilbert to Frege, 1/6/1900)

Where Frege parts company with Hilbert is where Frege parts company with modern logic and metatheory. And he wasn’t necessarily wrong. Frege believed that axioms were facts that followed from concepts we already knew. Or, in other words, axioms expressed fundamental facts of intuition. Entailed in this is the assumption that we can have intuitive knowledge of our logical, and therefore mathematical truths. So, for Frege, axioms have explicit meaning. Accordingly, he believed it impossible to substantiate logical concepts under Hilbert’s axiomatic system since the only requirement was consistency. Hilbert states, “If the arbitrarily given axioms do not contradict each other with all of their consequences, then they are true and the things defined by the axioms exist. This is for me the criterion of truth and existence.” (Letter, Hilbert to Frege, 12/29/1899) Yet, Frege’s point, which is deeply pertinent, questions whether Hilbert’s notion of consistency necessarily implies existence.

To accentuate his worry, Frege put forth the following example to Hilbert:

Suppose we know that the propositions

1.A is an intelligent being

2.A is omnipresent

3.A is omnipotent

Together with their consequences did not contradict one another, could we infer from this that there was an omnipotent, omnipresent intelligent being? (Letter, Frege to Hilbert, 6-1-1900)

On the contrary, Hilbert , as a Formalist, saw no use for intuition when it came to the actual validation of axioms since his system treated definitions as higher level concepts who’s truths were not self evident. His was a system in which points, lines and planes could be anything one liked. In other words, the logic appropriate for any one examination is derived a posteriori from the particulars of that examination. Yet, if Hilbert is to defend this position, and that consistency is enough to qualify existence, then surely some restrictions on his mathematical system must ensue. Or, as absurd as it may sound, do we just take mathematical existence to fall short of actual existence? Doing so would mean taking the proposition “there exists an X such that Px” to mean “It is consistent to assume that Px for some X”. As a response, Hilbert conjectured that all consistent axiom systems have a model which, therefore, makes an appeal to intuition unnecessary in regards to spatial or mathematical reasoning. But, Frege disparaged Hilbert for wanting to “detach geomoetry entirely from spatial intuition…” (Letter, Frege to Hilbert, 1/6/1890) For Hilbert, however, intuition could never be a clear guide to independence since intuition is incapable of securing certainty. As such, from his perspective, it is only through the lack of contradiction that one could imply existence.

Hilbert further believed that his axioms provided us with mathematical concepts that were merely suggestive in use. He states, “I do not want to assume anything as known in advance…” As such, when discussing primitive terms (point, line, between) Hilbert chose to interpret them as unknown objects in an equation. This was due to the fact he was not concerned with whether or not the objects specified by his axioms, but rather whether or not his axioms characterized the objects that fit their description. He urged, “it would take a very large amount of ill will to want to apply the more subtle propositions of plane geometry or of Maxwell’s theory of electricity to other appearances than the ones for which they were meant…” (Letter, Hilbert to Frege, 12/29/1899)

Yet, in spite of all of this, there is one important aspect of the Frege Hilbert debate that is often overlooked. As we have discussed, it would be easy to dismiss Frege as being either blinded by or naive to Hilbert’s concepts. But this assertion is far too simplistic. For Frege painstakingly documented, from his point of view, how an axiomatic system could be understood without the requirement of our axioms being partially interpreted or possibly false. Moreover, he believed that it was possible to show both independence and the existence of non-Euclidian geometry without relying on uninterpreted statements. It is this achievement that makes Frege as the founder of the methods we still use today to prove positive consequence result. As such, for Frege, logic is a universal system of reasoning- a system that one reasons in and not about. And it is this difference that ultimately separates his conceptions from those of Hilbert.

From an epistemological view then, Frege saw different references to ‘point’ and ‘line’ not as uninterpreted, but rather as first and second level concepts; whereas the Euclidian geometry is a first level concept and the non-Euclidian is a second level concept. He states, “within which, aside from the Euclidian point-concept, still other concepts fall…this second-level concept must also be a completely determinate one, but it behaves toward the first-level objects falling within it in a way similar to that in which a first-level concept behaves towards the objects falling under it.” (Frege, On the Foundations of Geometry, Pg 68) So, Frege’s rejection of an uninterpreted language does not entirely rule out the existence of modern geometries. It does, however, pose a problem for establishing the truth of its second order axioms since we cannot have a spatial intuition beyond the planer Euclidian space. In his second letter to Hilbert he complains of this very thing. “A logical danger lies in the fact that you say, for example, ‘the axiom of parallels’, as if it were the same in every other geometry. Only the wording is the same; the thought-content is different in each particular geometry….The characteristics which you state in your axioms undoubtedly are all of a higher level than the first.” (Letter, Frege to Hilbert, 1900) So, his point here is to rehabilitate Hilbert’s project in a way that would reject the use of pseudo-propositions and undefined terms, not to abandon it altogether. Subsequently, when viewed through this lens, Hilbert’s axioms do not define the concepts of ‘point’, ‘line’, or ‘betweeness’, but rather they define a second order concept of a three-dimensional Euclidian space.

So, from this, perhaps we can derive an account of their correspondence in which Frege does not dismiss Hilbert’s work entirely, but rather Hilbert’s own account of it. Frege states, “Neither the axiom nor the propositions that follow have a sense of their own; rather, the axiom is an antecedent pseudo-proposition and these propositions that follow are consequent pseudo-propositions which form one or several real propositions whose parts they are.” (Frege, On the Foundations of Geometry, Pg 77) Thus, the axioms are part of a general theorem that has a sense, even if the smaller parts do not. And this would mean that Frege was not blinded to the intricacies of Hilbert’s work, but rather unaccepting of what he perceived as its surface level confusions. Frege was deeply attached to a calculus of pure thought and did not want to see it morphed into a calculus of mere symbols. At the end of the day, it was this standard that Frege sought to uphold. Yet- at what cost

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