On Philosophy

November 8, 2007

Weiner And Jeshion On Frege

Filed under: Logic — Peter @ 12:00 am

Previously I discussed two possible understandings of the motivation behind Frege’s project, what it was attempting to accomplish, as developed by Benacerraf And Kitcher. Benacerraf proposed that Frege’s project was primarily a mathematical one, and that the philosophical trappings surrounding it were of little importance to Frege. The problem with this interpretation is that Frege himself doesn’t seem to view his project in this light. While it is clear that his motivation was partly mathematical it doesn’t explain his entire project, why he wrote Foundations, and why he was so concerned with refuting alternative definitions of number. Kitcher interprets Frege’s project as not standing by itself, but as a kind of correction to that of Kant, which ties to improve the epistemological status of arithmetic, both by providing it with more rigorous proofs and by showing that it is actually analytic, which frees us from depending on pure intuitions about numbers to justify its fundamental axioms, a practice that Frege sees as questionable. The problem with this interpretation of Frege’s motivation is that it understands him as seeking to make the theorems of arithmetic more certain, but the way in which Frege actually establishes his reduction of arithmetic to logic essentially implies, under this interpretation of him, that we can only be as sure of the correctness of this reduction as we are of the theorems of arithmetic independently of the reduction, which means that Frege’s project cannot actually provide us with more certainty about arithmetical truths. If we are to understand the purpose of Frege’s project we must, therefore, consider other interpretations of its significance, and so we turn to the work of Weiner and Jeshion, who also take up the task of explaining what Frege saw as the significance of his own work.

Like Kitcher, Weiner, in her paper “The Philosopher Behind The Last Logicist”, attempts to explain Frege’s motivation as primarily epistemological. However, Weiner does not see Frege as attempting to improve our epistemological position with respect to numbers, rather she understands him as trying to understand it, to explain how it is, exactly, that we come to know facts about numbers. Frege then is after the sources of our knowledge, at least when it comes to arithmetic. This then is why the analytic/synthetic distinction is emphasized by Frege in Foundations, not because he is trying to build on Kant, but because Frege has defined these terms via reference to the grounds of proofs, and thus to establish whether arithmetic is analytic or synthetic establishes what it is proved from, and thus where knowledge of it ultimately springs from.

However, this interpretation of Frege’s project runs into problems when we consider how Frege gets form his logical axioms to the theorems of artithmatic; he must introduce definitions of the number zero, what it means to be a number in general, the succession of numbers, and so on. The problem is that these definitions are not necessitated by his logical foundation, indeed they can’t be because they contain terms that are not included in that foundation. Thus it would seem that any theorems of arithmetic yielded by this system are grounded not only on the logical laws, but on these definitions as well. And the definitions are argued for by Frege in essentially a philosophical way, so it would seem that the definitions are themselves justified by our existing understanding of number in some fashion. Thus Frege’s project, under this interpretation, would seem circular, because, in seeking the grounds of knowledge for arithmetic (truths about numbers) he comes to the logical axioms and these definitions, which are grounded in our knowledge about numbers. And thus Frege’s project would have revealed essentially that the arithmetic theorems it yields are grounded in our knowledge of numbers, which shows absolutely nothing about the sources of this knowledge.

Weiner recognizes this problem, and suggests that Frege’s definitions are not actually justified by our understanding of what numbers are. Indeed they are not epistemologically justified at all. What Frege is attempting to do, according to Weiner, is not to repair or improve our existing conception of arithmetic, but instead replace it with a new systematic science, which can yield theorems analogous to those of arithmetic, and thus which can effectively replace it. In this new systematic science the definitions of zero, number, and so on are not justified, but are rather simply stipulative abbreviations for more complicated constructions, which happen to prove fruitful by yielding a number of interesting formulas.

Such an understanding of Frege’s project escapes the apparent epistemological circle, similar to the one facing Kitcher, namely that the correctness of Frege’s reduction seems to depend on the correctness of our existing understanding of arithmetic, by arguing that Frege is replacing arithmetic with something new. But this raises serious issues for understanding Frege’s motivation, because it undercuts the motivation that we began this very account with. If Frege’s motivation was to understand how we can know the truths of arithmetic he will have failed on this account, because what he has done is created a new system, arithmetic(F), which replaces our less rigorous arithmetic(O), and thus has explained only how we know the truths of arithmetic(F), but not those of arithmetic(O). And it was how we know the truths of arithmetic(O) that was the interesting question, because when we create a new system, essentially from scratch, there is no question about how we come to know things about it. Even more problematic for this understanding of Frege’s motivation is that there is no reason to replace arithmetic as formalized with the Dedekind/Peano axioms with Frege’s system, because to be an adequate replacement it must agree completely with the Dedekind/Peano axiomatized arithmetic. But, given that it agrees perfectly with it, there is no point in replacing that system with Frege’s, because the fact that Frege’s system is defined in terms of logic conveys no intrinsic advantage. And thus Frege’s project would appear to be completely without motivation, understood in this way.

Thus we come to our final interpretation of Frege’s motivation, as explained by Jeshion in her paper “Frege’s Notions of Self-Evidence”. Jeshion claims that Frege’s primary motivation is to prove the theorems of arithmetic, which are themselves not self-evident, from axioms that are. In doing so we become more confident of the theorems justified in this way, and thus Frege’s project conveys epistemological benefits. This might seem to make Jeshion’s interpretation fall prey to some of the same objections that faced Kitcher’s, that Frege’s project seems to presuppose the truth of certain facts about arithmetic, and Weiner’s, that in getting to the truths of arithmetic from logic we are forced to employ definitions that are not themselves logical truths. But Jeshion’s interpretation has ways of avoiding these problems.

First, Jeshion does not understand Frege as taking claims that are self-evident to be beyond any possible shadow of doubt. Thus Frege’s project is not meant to establish the truths of arithmetic with a perfect certainty where before they might be doubted. Instead a proof from self-evident truths is only intended to improve our epistemological position. The idea, I suppose, is that by producing such a proof of a proposition that was thought to be true we increase our certainty, because of the certainty we have in each of the self-evident links in that chain. However, no self-evident proposition is immune from being overturned, and things that were thought to be self-evident may later turn out to be in need of proof or analysis themselves. But this still leaves us with the problematic definitions, which are key components in this chain. But Jeshion has an explanation for this as well, she claims that the definitions too may have the property of being self-evident, and indeed any proper definition, in which the senses of the terms on both sides is the same, is. Thus Frege’s defense of his definitions is explained, not as an attempt to prove them, but as an attempt to explain them so that their self-evidence can become apparent.

But this interpretation of Frege has its problems as well, some of which are pointed out by Weiner in her paper “”What Was Frege Trying to Prove? A Response to Jeshion”. The central problem that this interpretation of Frege faces is that it leaves unexamined why an axiomatization of arithmetic, such as Dedekind/Peano, wouldn’t have satisfied his demands of self-evidence. Frege does claim that certain propositions of arithmetic, such as sums involving large numbers, seem in need of proof, but he does not say the same about the associative law, for example. He indicates that it would be preferable were a proof of it to be found, but he does not say that such a proof is necessary. And, more pressingly, if this is indeed how Frege saw his project, it would be utterly mysterious why he was so concerned with proving the fundamental axioms of arithmetic, but had no problem accepting those of geometry. Certainly it is hard to argue that the axioms of geometry are somehow more self-evident than those of arithmetic, and less in need of a proof. Additionally, when discussing Cantor’s system he does not criticize it for being based on propositions which aren’t self-evident, but rather claims that it is lacking because precise definitions of number and following in succession were not given and proofs not provided that follow from those definitions.

This essentially exhausts the available literature on Frege’s motivation, and yet we have not yet come to a satisfactory understanding of it, and thus of why Frege’s project should matter to us. I think the problem lies in seeing Frege’s work primarily in the light of the way in which it contributed to later mathematics, and thus our attention is drawn to Frege’s project primarily as one of proving things. But there are numerous indications that what Frege was primarily interested in was defining numbers, and that reduction of arithmetic to logic was primarily a vindication of these definitions. Explaining Frege’s project in these terms is a task I will turn to tomorrow.

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