Mathematical Knowledge: Where Truth and Proof sit in the Analytic Tradition Over the course of the semester we have been introduced to a variety of philosophers who produced a great portion of their work in the analytic tradition of philosophy. We focused on specific works from Bertrand Russell, A.J. Ayer, Ludwig Wittgenstein, and Wilfrid Sellars. The project that I am embarking on can be extended to the authors we read in this course but didn't focus on; indeed, there are some interesting things to point out about their positions on knowledge and analytic truth, which we don't fully see in these four philosophers I will be focusing on. For some, such as Mill or Frege, the examples are obvious if anyone has studied the responses to their work. For others, such as Quine or Carnap, it perhaps isn't so obvious; indeed, my examples against them are far less 'set in stone' and far more exploratory. But for the sake of brevity and clarity, I will focus on 'the big four'. The primary books by these authors that we've looked at are 'The Philosophy of Logical Atomism' by Bertrand Russell, 'Language, Truth, and Logic' by A.J. Ayer, 'On Certainty' by Ludwig Wittgenstein, and 'Empiricism and Philosophy of Mind' by Wilfrid Sellars. While these authors seem almost overwhelmingly interested in issues relating to empirical truth, they have comments on what it means for something to be analytically true. This is true especially when they begin making comments about such things as mathematical truths, and these comments offer an entirely different and perhaps frequently overlooked avenue by which one can question, probe, and prod the espoused viewpoints. In this paper, I will assume at least a moderate level of familiarity with the works of each author. Indeed, for someone like Russell who made a name for himself in part for his contributions to philosophy of mathematics, it would be almost redundant to delve into his work in a rigorous way. Instead, I will attempt to offer examples to the philosophies argued for by the authors from the field of mathematics, to clarify what might be called gaps in their work. If anyone has ever studied the philosophy of mathematics, Bertrand Russell is perhaps the first name they would come across. Russell's project was in some ways an extension of Frege's. His major motivation was to resolve his now famous Russell's Paradox that arose from Frege's work, and so the Russellian position can be properly characterized as logicism: an attempt to reduce arithmetic truths to logical truths. Russell's development of class theory is perhaps one of his greatest achievements in the field. However, it can also point out to us what might be called major issues with his other work. In grounding mathematical truths in logical truths, Russell was attempting to demonstrate that mathematical truths were analytic ones. That is to say, they were true just in virtue of their logical form. In developing type theory, Russell found that the paradoxes he pointed out to Frege about impredicative definitions were also proving difficult for his own work. As a result, Russell introduced a principle of reducibility, which allowed him to produce a class which was predicative for any class. This principle is almost entirely necessary for producing anything beyond basic, natural number arithmetic. By this token, one must ask how it is that the principle is justified. Indeed, given the development of type theory, it seems ad hoc, instituted only to solve problems revealed after the fact. Indeed, it isn't knowable a priori, and it cannot be grounded in logic - this much Russell admits of the principle. However, a problem exists beyond this flaw. If one considers numbers to be certain classes, even predicative ones, then the levels of classes become almost cumbersome to navigate - one can easily get lost in the hierarchy. For the Russellian development, negative integers are, in one sense, simply relations on their positive counterparts. Herein lies the issue: any number beyond natural numbers in the usual construction is simply a relation, and not in itself a class of classes (as the naturals are). Indeed, '+2' is wholly distinct from '2'; yet this seems wildly difficult to accept, on at least a common-sense understanding of numbers. The difficulties presented by type theory and class hierarchies makes mathematics unwieldy and complicated, and the principles we have to invoke to properly do so without contradiction seem to make mathematics, in part, not wholly analytic, or at least the ways in which we talk about it isn't. In itself, the logicist project is a worthwhile consideration. But the problems presented to us by type theory make doing mathematics on any level from arithmetic to real analysis almost entirely unbearable. Indeed, it is unclear exactly how Russell might aim to reduce something beyond real analysis to logic and speak of it in wholly logical terms. Jumping ship from Russell's logicism we arrive at what might be considered an extension of the view: logical positivism. At the very least, the logical positivists attempted to ground analytic truth in facts about meaning, rather than a wholly logical position like Russell. For Ayer, an analytic truth would be one in which its truths depend entirely on the meanings of the symbols it uses, and for Ayer this is all that mathematical truths amount to. This position seems rather enticing. Indeed, mathematical truths exist outside of our experiences of fact, and are not verified by some sort of observation of the world. Instead, mathematical truths exist wholly separately from proper experiences, and are true in virtue of their meanings alone. This move eliminates many objections to Russell's position on the matter. No longer do we have types and classes and relations to worry about. Instead, we need only focus on the meanings of our words, and the symbols we employ. As such, knowledge of mathematical propositions is simply knowledge of the correct use of mathematical language. But this identification of knowledge of mathematical propositions with knowledge of its use seems quite problematic. If this is the case, it seems entirely unlikely that there would be many unsolved problems in mathematics at all. There are a few cases for why these problems might be unsolved. For instance, it might be that they aren't mathematical propositions. Alternatively, it could be the case that we don't know what we mean when we use these propositions; we don't have a proper understanding of their meaning. However, both seem wholly unlikely. The Riemann Hypothesis [1] or the Golbach Conjecture [2] seem very much like mathematical propositions: they are written in the language of mathematics, they have mathematical consequences, and they wholly arise out of mathematical facts. It also seems unlikely that mathematicians don't know what it would mean for something like the Riemann Hypothesis to be true. Indeed, we know that if it is the case that it's true, then the Continuum Hypothesis [3] is true. As it turns out, the positivist tradition put forward by Ayer does not provide us a method by which we can make determinations of the truth of mathematical propositions. There is no algorithm by which we can check their validity. The fact that this is the case for such simple statements about natural numbers and primes in Golbach's conjecture is a proof of fact of this. Indeed, Fermat's Last Theorem [4] was not solved until only recently, yet anyone who reads the statement of the theorem knows exactly what it means - it is a statement simply about addition, integers, and natural numbers; some of the simplest things in mathematics. However, the mathematics used to prove its truth extend far beyond what any nonmathematician (and even many mathematicians themselves) would ever know. From this, it seems difficult for the positivist to make claims about the analyticity of mathematics in virtue of meaning alone. Wittgenstein might be the first author we read this semester who both simultaneously fails to provide a particular system which accommodates mathematical truths and formally accepts mathematical truths. The trouble with Wittgenstein is that he is not to be regarded as a mathematician, classically understood. Wittgenstein's philosophy of 'meaning as use' and things being descriptive of a language game as part of logic entail that classical mathematics might not be the only system of mathematical formalism. Indeed, Wittgenstein himself was influenced from lectures he attended by L.E.J. Brouwer, a famous mathematician and philosopher. Brouwer led the development of intuitionist mathematics, which sought to do mathematics in an intuitionistic logic, diverging starkly from the usual method of classical logic which we see employed by those earlier authors like Russell, Frege, or Ayer. For the intuitionist, a proposition A is true just in case there is a proof of A; the negation of A is true just in case there is a proof of not-A. In this way, the intuitionist does away with the law of excluded middle being a tautology, something that most modern-day mathematicians would not accept. However, it would be quite incorrect to say that Wittgenstein himself was an intuitionist. From what I have read, it isn't entirely clear what sort of logic Wittgenstein would have preferred. That being said, there are certainly interesting points offered by mathematics that allow us to perhaps better understand what Wittgenstein's theory of language games would produce for us. Take for instance, the relation 'is less than' [5]. One of the most fundamental things one learns about numbers is that 1 < 2, 2 < 3, etc. The idea that there is an ordering - indeed, a total-ordering - on numbers is oftentimes taken for granted. We can extend this notion from real numbers to things like the integers. After introducing zero, we can say things like -2 < 0 and 0 < 27, therefore -2 < 27. We can further generalize the notion of <, by saying that if a < c, a + b < c + b, given the proper framework surrounding addition. What's curious to note is that, for almost any 'normal' language user - someone who would use mathematical facts in a layman sort of way - these points are almost trivial. But take for instance, the complex numbers. Those numbers which have the form a + bi, where a and b are real numbers and i is the square root of negative one. Suppose you were to ask someone whether 1 + 2i < 3 + 4i was the case; what would they say? Perhaps they would be inclined to point out that, because 3 and 4 are larger than 1 and 2, it seems clear that this statement is true. On the face of it, it's unclear how they might be wrong. But consider the following: is i less than or greater than zero? This is an almost elementary exercise for those who pursue a degree in mathematics - given either assumption, there will always be a contradiction [6]. As it turns out, this is a demonstration that the complex numbers are not an ordered field like the natural numbers, the integers, or the real numbers. There are other 'trivial' facts that likewise fail for larger number systems. Division in the complex field works differently than it does for real numbers. One of the more humorous things to note is that 'ordinary' numbers cannot describe rotations; indeed, appeals to complex numbers must be made to rotate things in two-dimensional space [7]. The question we are forced to ask ourselves given these revelations is, who was right? Given a language game, it seems that mathematicians know something different about the ordering relation than the layman does and therefore is perhaps using it differently, and is thus speaking a different language; they could be said to be using the relation differently. Yet we continue understanding ourselves and each other with no real issues arising from it, as if we weren't saying anything different at all [8]. The point at the very least is an interesting one, and while it isn't necessarily a strike against the 'meaning as use' idea itself, what we once considered to be immutable and a priori truths certainly shed a new light on the sorts of problems one might face in using a Wittgensteinian framework. From this avenue, we arrive at the Sellarsian picture of knowledge. The question of whether someone can be said to have knowledge of mathematical truths on Sellars' view is an interesting one. Due to its close relation to Wittgenstein's philosophy about meaning and the interrelation of concepts, Sellars seems to avoid any sort of serious qualms one might have with the idea of rule following giving rise to knowledge. The crux of the point here is that within the space of mathematical truths, the question of whether it is analytic or not is unnecessary. As these beliefs sit in a logical space, the premier question is whether there is a logical justification for a certain belief or fact that is known. What this means for something like the prior example of relations on numbers is that if a person can justify their claim, they can be said to have knowledge. As a result, both individuals - those who are aware of the relation as it applies to real numbers and those who realize how it doesn't apply to complex numbers - can be said to have mathematical knowledge. As a result, it appears that the only thing required of someone to have mathematical knowledge is that they are able to provide some sort of demonstration that it is the case; in general, this might be called a proof. The only foreseeable problem to arise from this is that individuals might be hard-pressed to prove basic mathematical truths in a general sense. We might be welcome to the idea that, if someone claims to know that 2 + 2 is in fact 4 and they can show this fact in some sort of way, they know it is the case that two and two make four. However, the question arises of whether we should demand such proofs for every claim about mathematical facts if they haven't demonstrated the principle in general. It seems that, at this point, the demand for justification becomes too extreme, as we might be tempted to ask everyone who espouses mathematical facts, even in passing, to become casual mathematicians. However, it seems that we should be more than allowed to waive such a requirement in general; our justificatory practices need only extend so far if we share common beliefs. In this way, Sellars has a position very similar to a sort of Wittgensteinian, language game holism he can pull from to deny this sort of line [9]. [1] The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part ½. The Riemann zeta function is itself in one part a way to study the distribution of prime numbers. [2] The Golbach conjecture states that every even integer greater than two can be expressed as the sum of two primes. It dates back to the 18th century and has yet to be solved. [3] The continuum hypothesis states that there is no set whose cardinality is strictly between that of the integers and the real numbers. That is to say, there is no set bigger than the integers that is also smaller than the number of real numbers. The set of real numbers is itself thought to have a cardinality equal to that of the integers as a power of two. [4] Fermat's Last Theorem states that there are no integers a, b, c such that an + bn = cn for integers n > 2. [5] From here on, this relation will be denoted by <, its opposite, >. [6] Let 0 < i. Then 0 < i2. Thus, 0 < -1, a contradiction. So it must be that i < 0. But then i4 = 1 < 0, a contradiction. [7] Even more strikingly, rotations in three-dimensional space require quaternions, we might think of them as 'four dimensional numbers'. [8] This all hints at a larger issue for Wittgenstein as it relates to rule following in general, and there are some excellent papers about it. One of the more interesting points is about the 'plus' function versus the 'quus' function, by Kripke. [9] A great deal of thanks for this is due to Stewart Shapiro's 'Thinking About Mathematics'. For those interested in reading more on this issue as it relates to realism and antirealism, Shapiro's work offers an excellent covering of these problems historically and contemporarily. ________________________________________________________________________________ Dilyn Corner (C) 2020-2022