On Philosophy

November 9, 2007

Frege’s Ontological Motivation

Filed under: Logic — Peter @ 12:00 am

Previously I have discussed four different ways of understanding the significance of Frege’s project. Two we might characterize as epistemological, as saying that Frege’s project is important because through it we improve our knowledge about arithmetic, in the sense of making it more certain, or that we are to uncover how we know about arithmetic through it. These interpretations of Frege’s motivation are undermined by the fact that his project simply can’t succeed if that is indeed what he is trying to achieve. Frege’s reduction relies on definitions of what numbers are, definitions that are validated by their ability to prove the accepted arithmetical truths. Because of this his project can’t improve the certainty of our knowledge since it’s correctness is justified by the very claims it would seem to establish, nor does it reveal how we come to know things about numbers, since the definitions themselves are not proved in a rigorous way. The other two interpretations we can characterize as mathematical, which explain Frege’s project as essentially trying to establish proofs for their own sake, with possible epistemological side effects as an additional benefit. The problem with the mathematical interpretations of Frege’s project is that it makes much of the Foundations, specifically the parts devoted to refuting certain conceptions of number, pointless in terms of the mathematical motivation of Frege’s project. And, secondly, it isn’t clear under this interpretation of Frege why he would be so intent on proving arithmetical truths, but would be content to essentially ignore geometric truths.

As I understand Frege he had two motivations for his project, not one, which often overlapped. The problem is, for modern readers, that one of Frege’s motivations is something we simply no longer care about, and so we overlook its importance for Frege. Specifically, I claim that Frege was partly motivated by a desire to vindicate his begriffsschrift, to demonstrate its mathematical value as a rigorous logical system in which all deductions are gap-free. Obviously this isn’t a motivation we can share, because we are fully convinced of the value of such systems when there is some question as to the validity of the proof, and we are equally aware that when we want to actually get somewhere in mathematics we must put them somewhat to one side because they make proofs a bit too complex for mere mortals to handle. But this is not something that Frege’s contemporaries understood. Frege had introduced his system and believed that using it would improve mathematics. But obviously no mathematician was simply going to pick it up and start doing mathematics with it without some reason to believe that they would benefit from it.

So, from Frege’s point of view, what was needed was a demonstration of the value of the begriffsschrift. One way to do that would be to prove some interesting new mathematical theorems with it. However, if Frege proved something new then questions about the validity of the proof and what exactly it shows would be entangled with questions about the usefulness of the begriffsschrift. And, while proving a set of already established truths would not run into this problem, at the same time it wouldn’t really show the value of the begriffsschrift, since they were already considered satisfactorily proven by other means. This is why proving the truths of arithmetic is useful, because, while they had not been satisfactorily proved before from logical principles, neither was there any question about their validity. Understanding Frege’s motives in this way explains why he places such a heavy emphasis on the desirability of proving the truths of arithmetic, because if we desire such proofs and Frege can provide them with his begriffsschrift, thus showing it to be useful in mathematics, even if we didn’t need it to validate those truths or to justify how certain we are of them.

We might understand this as Frege’s pragmatic motivation, but it is also a mathematical motivation, and leaves many of the same questions open as the mathematical motivations discussed previously. For example, why didn’t Frege instead attempt to prove the truths are arithmetic from what he calls Hume’s principle, which emerges as an intermediate step in the proofs given in Basic Laws? Surely proofs from this principle would still show that the begriffsschrift is useful. And certainly disproving what other people think about numbers doesn’t help validate it at all. Additionally, we must also wonder why he thought that proving the truths of arithmetic was possible and possibly beneficial when those of geometry were allowed to stand without any defense. Certainly Frege indicates that his choice was more than arbitrary.

To answer these questions we must appeal to Frege’s motivation, which I describe as ontological, to understand what numbers are. This might seem as absurd way to interpret Frege to some, but certainly Frege’s project can be seen as a reduction of arithmetic to logic. Now consider what a reduction of the mind to the brain shows, it shows that the mind really is the brain, an ontological claim, and not a claim about how we come to know the mind or on what basis facts about it are justified. Similarly, it is thus reasonable to assume that a reduction of arithmetic to logic tells us what arithmetic, and thus numbers, are, as well as whatever else it does. But you don’t have to take my word that this is Frege’s motivation, he begins his introduction by considering answers that might be given to the question “what is a number?” and why they are unsatisfactory, and this strongly indicates that this question was at least one of the problems he was directing his attention towards.

But perhaps the best evidence that Frege had the ontological motivation I attribute to him is simply they way in which it dissolves many of the questions that the way Frege proceeds in his project raises. For example, if we grant Frege an ontological motivation then the reason that much of Foundations is devoted to arguing against alternate definitions of number is obvious; Frege wants to know what numbers are and finds these other definitions unsatisfactory. Moreover, if someone accepts them it will clearly prevent them from coming to an adequate understanding of what numbers really are, and thus for Frege’s ontological goals to be accomplished he must first show that what numbers are hasn’t been satisfactorily addressed in some other way. It also explains why Frege was not satisfied with Hume’s principle as a foundation for arithmetic. The problem is, essentially, that while it says when numbers are equal it does not say what numbers are, it does not define them. And since Frege is after such definitions it will not satisfy him, even if he could have proved all the truths of arithmetic from it. And this ontological motivation explains, in conjunction with some of his comments about geometry, why he was not similarly interested in proving those truths. Frege thinks that the objects of geometry are given to us in our pure intuitions of space, and thus that, as they are given to us, we already know what the objects of geometry are. Thus there are no significant ontological questions to answer of the kind that might provide primitive definitions to construct proofs of the axioms from. Finally, Frege’s early comments about the analytic/synthetic distinction are put into a proper perspective, not as part of an epistemological structure, but as a way of discovering what numbers are. Because, according to Frege’s understandings of the terms, whether a statement is analytic or synthetic depends on the ultimate foundation from which it can be proved. But that ultimate foundation will always include definitions of the terms, unless the terms are primitive, and are themselves the foundation. Thus if Frege can show that the truths of arithmetic are analytic he will have shown that the definitions of number are logical definitions, and thus that numbers are logical entities, partially addressing his ontological concerns.

However, while this provides an answer to many of our superficial questions surrounding Frege’s project, we must also ask whether his definitions and the way in which he arrives at them can be reconciled with an ontological motivation, or whether they only make sense as a mathematical enterprise. In the later sections of Foundations Frege essentially proceeds to arrive at his definitions by a kind of conceptual analysis of the way we use numbers, considering various possibilities and eventually arriving at his precise definitions. After constructing them he states that the definitions must show their worth by proving fruitful, specifically in this case by leading to proofs of the “well-known properties of numbers”, which some attention to is given in Foundations, and which is the primary focus of Basic Laws. But does this work as a way of discovering what the numbers are, or is it simply a roundabout way of arriving at mathematical definitions that are essentially stipulative?

Well, let us consider what a challenge to Frege on this point would entail, assuming that we are claiming that he has not arrived at the nature of numbers, and thus a genuine solution to the ontological question, by this procedure. Certainly we can’t claim that numbers are anything by logical entities, assuming we are convinced by his earlier arguments, which Frege takes to demonstrate that the arithmetical truths must be analytic (or at least give us good reason for believing them to be). Nor can we claim that numbers are indefinable, simply because Frege has provided at least a possible candidate for such a definition. So, if we are to disagree with Frege it must because we define numbers in some alternate way, but which is also in logical terms. Which means that we are conceding to Frege at least part of his ontological results, that numbers are logical entities. But even Frege himself admits that other definitions of numbers might be given in logical terms that also can be used to prove all the arithmetical truths, and this might seem to make any specific ontological claims of his unsuccessful. However, I think Frege would argue, confronted with such an alternative, that his definition is the best fit for the sense we already give to numbers, as is shown by the conceptual analysis leading up to the definitions. And so, unless we can produce a better definition of the numbers, we must admit that Frege’s project can be understood as revealing what numbers are, assuming we don’t find issue with the way in which we have reached this conclusion.

Thus this interpretation of Frege’s project and his motivation has three merits. First, it answers a number of questions regarding the details of why Frege proceeded as he did. Secondly, it gives the project an interpretation in which it can actually be successful in its own terms, as compared to the epistemological interpretations which would imply that the project couldn’t work even if it was error free, and that seems to be underestimating Frege’s intelligence. And, finally, it gives the work a significance that actually matters to us; Frege’s project is not simply a mathematical exercise using an inconsistent logical system which we can ignore, it attempts to answer a question that we might still address ourselves to, and one which is general enough to concern everyone, since it does not require us to buy into any of Frege’s specific assumptions to be concerned with it.


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