Defending Mathematical Realism

and the concrete was claimed to be epistemological. This difference is recognized throughout the history of philosophy in the distinctions made between the a priori and a posteriori as well as between the analytic and synthetic. The exact meanings of the terms are disputed, and sometimes the terms are rejected outright, but they (very) roughly convey distinctions that can be read into the difference between concrete and abstract objects. If anything is said to be known a priori, it means that there does not have to be something that causes the knowledge of that thing in the sense of causal used in the antirealist challenge. Analytic knowledge is derived from conceptual analysis or is definitional. Again, it seems as though there is no direct cause for analytic knowledge. The neo-Fregean works off a belief in the a priori status of mathematics in claiming that the demand for causal links to mathematical knowledge is inappropriate. HP gives the sense that the grasp of number can be prior to the grasp of the 1-1 relation corresponding to a number. Further, the 1-1 relation is not necessarily among physical concepts. The concepts used to construct the natural numbers are purely logical. Logic and other abstract concepts have this a priori status as well. With regard to number, it is not clear what causes the stipulation of HP, however. Is it the case that someone could come up with HP prior to experience or did they witness phenomena that needed to be counted and proceeded to lay down the definition. This question misses the crux of the neo-Fregean program guided by context principle. Neither option captures the situation correctly. It is not legitimate to consider a term in isolation, but only in context. What this establishes, is that the question, “what is number,” is invalid. There needs to be a contextual construal of how such objectreferring singular terms feature syntactically in true statements. It should not be asked what knowledge there is of books or of numbers, but what knowledge there is of the propositions in which their associated singular terms occur. The strategy embraced is to see what function „number‟ plays with regard to the meaning of the statements in which it occurs. By using the context principle, knowledge of number is just the same knowledge of the statements which feature numbers essentially. Knowledge of mathematical statements rests in their deducibility from certain axioms. The means for evaluating the truth of the statements is logic and analysis of the axioms. If HP can be accepted as a non-logical axiom, then mathematical knowledge is unproblematic. Because HP is non-logical, the neo-Fregean cannot be a pure logicist as Frege was. Instead mathematics is said to be analytic, where the truths of the statements are seen as deductions from the definition of number, HP, and second-order logic. It is because the truths of mathematics can be traced back in this way that there is no causal element required, so long as HP is an acceptable axiom. Mathematical knowledge, then, really can be asserted to be justified true belief. It is not clear what exactly the justification is for mathematical statements. There is computation which can be used to verify specific problems, and there is proof used to verify general propositions. The two are inferentially related, but not the same. Computation seems to be as reliable a process we can have. Proof causes some difficulties however, which Gödel‟s incompleteness theorems bring to light. They show that not the case that every arithmetical truth can be traced back to the axioms, that the truths are not recursively axiomatizable. Gödel showed that arithmetic is incomplete at its essence, meaning that the incompleteness does not depend on a particular axiomatization of arithmetic. He showed that there exist true but unprovable statements in any reasonable arithmetic; really, he showed that no set of axioms that describe arithmetic can be both complete and consistent. The incompleteness theorems are aimed at the formalization of mathematics as undertaken by Hilbert. To Gödel, the result meant that in order to know the truths of mathematics, we must have a mathematical intuition which gives insight into the facts otherwise underdetermined by the formalized system. Logicism, especially the near logicism of the neo-Fregeans, is not to be equated with Hilbert‟s formalism. Hilbert‟s system is an uninterpreted and otherwise meaningless mathematics. Logicism, by invoking HP, does not leave mathematics uninterpreted. HP may be seen as a logicist analog to Gödel‟s mathematical intuition. Whereas Gödel thinks that certain mathematical results require the supplement of intuition, the neo-Fregean claims that each result is analytic and therefore potentially knowable without invoking some vague faculty for recognizing mathematical truth. The neo-Fregean stops just short of claiming that each result is a logical consequence. Hilbert sought an absolute finitary consistency proof of arithmetic, but the neo-Fregeans admit just relative consistency. We can only say that HP plus second-order logic is probably consistent and that the problems that Gödel‟s theorems present are just on a par with concerns regarding the use of second-order logic. The force of Gödel‟s theorem comes from the fact that firstorder logic is complete and arithmetic is not, which implicates arithmetical axiomatization as the culprit. If logic is extended to higher orders, then it is no longer clear that arithmetic is the problem, and instead the problem might be that logical truth is as difficult to account for as arithmetical truth. This can be taken as a reason why higher order logic is only tenuously referred to as logic. This is not a good situation either, but it highlights the differences between the consequences of Gödel‟s incompleteness theorems for logicism and formalism, in that incompleteness is not an especial problem for logicism. The problem shifts to justifying the use of second-order logic with HP to derive arithmetic. The problems with second-order logic crop up again in the “Bad 28 HP is consistent with second order logic, that is. Company” objections to using HP, and hopefully there, it will be successfully argued that second-order logic is not especially problematic. To give a quick response here, it might be said that even though arithmetic is incomplete, the addition of HP and the realist attitude implicit in it have the ability to resolve any of the questions which are mathematically significant. Hale and Wright are unconcerned by incompleteness. In the introduction to The Reason’s Proper Study, they claim that by merely seeking to obtain the foundations of arithmetic rather than attempting to show that every mathematical truth is derivable as a theorem they avoid “an obvious clash.”Their claim is even more basic than the one above. All they hope for is to show that PA or an equally good set of arithmetical axioms can be derived from HP and second-order logic. That this is proved by Frege‟s theorem means that as long as HP can be accepted, all is well. If PA is incomplete, this is a mathematical problem, not a philosophical one. I am not sure that it is so simple. If mathematical truth is to be displayed through proof, and there exist true but unprovable statements, a gap appears. There is the option of saying that the truth of the statements, while unprovable, are subject to conceptual analysis to a degree that they can be rendered analytic. It seems this is the best option to take for the neo-Fregean. The program already relies on a view of mathematic being analytic. There may be room to engage in definitional manipulation outside of logic that would make the truths knowable. Also, there is the possibility of laying down new definitions by a process like HP to uncover further truth. D. Further Considerations 29 Hale and Wright (2001: 4, fn. 5) The first order of business that must be handled is the Caesar problem. Recall that the inability to determine the truth or falsity of propositions such as „2=Caesar‟ pushed Frege to make the ill-fated move of stipulating the inconsistent BLV. It is necessary that if we are said to have knowledge of numbers then we should be able to evaluate whether 2 and Caesar are the same thing or not. HP does not provide insight into the matter. It provides identity criteria among numbers, but does not provide general identity criteria for numbers among the universe of objects. To resolve the Caesar problem, Wright introduces a supplement to HP. N: Gx is a sortal concept under which numbers fall only if there are, or could be, singular terms „a‟ and „b‟ purporting to denote instances of Gx such that the truth-conditions of „a=b‟ could adequately be explained as those of some statement to the effect that a 1-1 correlation obtains between a pair of concepts. The supplementary principle makes explicit the somewhat obvious claim that identity can only hold among things which can be situated under the same identity criteria. Numbers cannot have one set of identity criteria in one context, and different set in the next. If a user understands Caesar, he must grasp that Caesar is a person, and thus subject to person identity conditions, of which we have not defined, and will not define, but that surely do not include 1-1 relations. By adopting the innocuous principle we can deal with the (absurd) Caesar proposition. Any attempt to identify Caesar with 2 through bijection is incoherent, and yet it is intrinsic to 2 that it can coherently be analyzed under such relations. It is a necessary property of any natural number, and so Caesar‟s failing to be relatable through the 1-1 relation precludes him from identifying with 2. 30 Wright (1983: 116) Up to this point, we have more or less accepted HP as true in virtue of an adequate explanation of the concept number, and its use justified. Considering the body of work regarding HP, this is not the case. The accept