On Philosophy

November 7, 2007

Benacerraf And Kitcher On Frege

Filed under: Logic — Peter @ 12:00 am

Throughout his professional career Frege was focused on proving the truths of arithmetic in terms of something simpler, at first logic, and later in terms of geometry, when paradoxes were uncovered within his logical project. But Frege’s motivation for this project remains obscure. Generally we can understand the motivation behind a project, and its importance, by understanding the implications it was thought to have by its creator. So we must ask ourselves, what would the ability to reduce arithmetic to logic imply? And what consequences does the apparent inability to consistently construct this reduction have? By answering these questions we will uncover Frege’s motivations. Of course I don’t want to imply that Frege had only a single motivation for his project, given the effort he spent on it I suspect that he had a number of reasons for pursuing it. What I am after then is a kind of primary motivation, one that can explain the entire project and which is consistent with the assumptions behind it. And, furthermore, it must be a motivation which is robust enough to survive the apparent inability to reduce arithmetic to logic given the problems with basic law V, such that it gives even the failure of the project implications. A simple interest in seeing whether arithmetic can be reduced to logic, for example, would not suffice given this second criterion, since it gives us no reason for caring about Frege’s project given that it was unsuccessful.

The problems inherent in uncovering the motivation behind Frege’s project are aggravated by the tendency of certain empiricists, who used Frege’s work for their own ends, to project their motivations onto him, as Benacerraf points out in his paper “Frege: The Last Logicist”. The empiricists saw the existence of synthetic a priori truths as contradictory to their project, because it would seem to imply that there are ways of arriving at knowledge about things, knowledge not contained in the ideas we use to describe them, without any necessary connection to experience. Thus if the empiricist project was to succeed these philosophers saw the need to explain all apparent cases of synthetic a priori truths as either analytic or a posteriori. Given this aim the reduction of arithmetic to logic seems eminently desirable, because it will show that the truths of arithmetic are really analytic truths, and, together with the reduction of the rest of mathematics to arithmetic, this will vindicate their project. But Frege certainly did not see his own work in this light. Although Frege originally thought that arithmetic was analytic he had no problem with synthetic a priori truths in general, and he refers to geometry as an example of such truths on a number of occasions. So, whatever Frege’s motivation was, it was certainly not to vindicate some kind of empiricist project.

To Benacerraf this indicates that our intuition that Frege had an epistemological goal in mind is flawed, and arises from illegitimately interpreting his project through the eyes of the empiricists. According to Benacerraf Frege’s project was primarily a mathematical one, and thus had primarily mathematical motivations. Frege’s motivation then stems from concerns with the foundations of arithmetic, and mathematics in general. Frege comments that mathematics is currently pursued in a haphazard fashion, with premises accepted simply because they have yet to be shown to give rise to any contradictions. He describes this as a merely empirical certainty, and implies that it should be obvious to us that mathematics should not be done in this way. This explains Frege’s motivation behind the introduction of his begriffsschrift (his logical notation), which will allow mathematicians to produce completely rigorous and gap-free proofs.

But that alone does not explain why Frege sets about to reduce arithmetic to logic. Of course we might argue that the axioms of arithmetic seem to need some justification, otherwise it might be possible for them to give rise to contradictions, as Frege sees it. But clearly this is not Frege’s actual motivation, for two reasons. First of all he doesn’t see any need to prove the geometrical axioms in this way, and it could equally well be claimed that we have no more than an empirical certainty that those axioms won’t give rise to contradictions. And, secondly, in the Basic Laws Frege validates his definitions of number in logical terms by showing that they satisfy the Dedekind/Peano axioms. If it was possible that those axioms were invalid then whether the proper logical definitions of number satisfy them or not would be irrelevant to whether they were properly definitions of number, because if they are possibly incorrect axioms then it is possible that the correct definitions of number don’t satisfy them. Benacerraf, avoiding these problems, suggests that Frege’s reduction of arithmetic to logic was simply a matter of mathematical curiosity. Like many mathematicians perhaps he wanted to see which further formulas followed from his logic, and thus extending it to include arithmetic was simply an exercise in examining the deductive power of his system. And certainly the fact that Frege discusses the analytic/synthetic and a priori/a posteriori distinctions solely in terms of justification may indicate that proofs were his primary concern.

While a compelling explanation of why Frege developed his begriffsschrift, this understanding of Frege’s motivation for reducing arithmetic to logic falls short in two ways. First it leaves why Frege would later try to reduce arithmetic to geometry completely mysterious, unless it is as a completely new project, unconnected from his previous work with logic. Secondly, it also makes Foundations seem completely superfluous. The work of actually proving arithmetic from logic is taken up in Basic Laws, and if that was what concerned Frege why write Foundations at all? Furthermore, Foundations is concerned not only with exploring issues surrounding Frege’s project, but with refuting other definitions of what numbers are as well. If Frege is motivated primarily by a desire to prove further facts from his logical system there would be no need for such arguments; certainly few other mathematicians, setting out to explore the consequences of some axioms, first argue that all other ways at arriving at the claims they seek to prove are illegitimate in some way. And the fact that Frege does spend this effort implies that his project is more than strictly mathematical.

Let us turn then to Kitcher’s explanation of Frege’s motivation in his paper “Frege’s Epistemology”. As is implied by the title Kitcher understands Frege’s motivation as primarily epistemological, as a search for how we can know facts about arithmetic. Thus Kitcher agrees with Benacerraf that Frege’s systematization of the subject was meant to give it a certain clarity and certainty that was missing in other treatments of arithmetic. But to explain why arithmetic is to be reduced to logic we must understand Frege as building on an existing epistemological foundation, namely that of Kant. Whether something is analytic or synthetic has significant implications in Kant’s epistemology, because the way we justify such claims, and the certainty we can have about them, differs between the two. What Frege can be understood as attempting then is to correct a small problem with Kant’s project, which classified arithmetical truths as synthetic a priori.

If that is indeed what Frege is doing then he must see some problem with the idea that arithmetic is synthetic a priori, and indeed Frege makes a number of remarks seemingly to this effect in Foundations, where he implies that there is something wrong with our intuitions about numbers, that somehow they are less clear than our geometrical intuitions. And this has the additional benefit of explaining why Frege then attempted to reduce arithmetic to geometry after he became convinced it couldn’t be reduced to logic, because if such truths must be synthetic a priori then the only way that they can be really justified, if indeed out pure intuitions about numbers are lacking, is to show that our arithmetical judgments are necessitated by our geometrical judgments, which are in fact properly grounded in intuition.

Again, there are two problems with interpreting Frege in this way. The first is purely textual, which is that Frege rarely mentions that analytic/synthetic distinction after Foundations. Clearly this isn’t necessarily devastating to this interpretation of Frege’s motivation, but it is puzzling, because why, if these epistemological foundations of arithmetic where what Frege was primarily concerned with, would he never mention them again? Secondly, we have the problem that arose while considering Benacerraf’s interpretation, namely that it seems inconsistent with the way that Frege actually justifies his definitions of number, by showing that he can prove the standard axioms with those definitions. Obviously if that is what justifies the correctness of our definitions then we can never be more certain of those definitions than we are of the axioms themselves. And so, if Frege’s motivation is epistemological, then his project must have been a failure even before the problems with basic law V were uncovered, because, given the way that Frege justifies his reduction of arithmetic, it can never be more certain than those axioms. But, given Frege’s remarks on the geometrical axioms in “Logic in Mathematics”, namely that in order to deny them that we must mean something different by words such as “line” and “parallel”, I think it is reasonable to assume that he did not see the axioms of arithmetic as the least bit uncertain. Which means that Frege’s project is probably not failing by his own standards on epistemological grounds, but, at the same time, it means that he must not be simply trying to make a small correction to Kant’s project as Kitcher claims.


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