Ethical Objectivity Through Mathematics

In Search of a Picture of Ethical Objectivity Through Mathematics Chris Harden If we afford the position that relativism about ethical truth is not a coherent position and fails to conform to how we actually live in the world, then we seem to be supposing the existence of some sort of objective, independent standard for understanding and assessing moral truth. The burden of proof is now on us to try and provide some account of what this ethical objectivity might look like and how one is to understand it as objective. The elusive part is the ambiguity in how people use the word objectivity. Many people, for instance, would grant things like mathematics, logic, and some of the sciences as instances or means of achieving, objectivity, knowledge, and truth. The goal of this paper will be to examine how it is that we have come to understand systems of thought like mathematics as objective and to see if there is something there that can be abstracted in an attempt to get a picture of what an objective system of ethics might look like. To be sure, I hope by examining the emergence of objectivity in mathematics and logic we may find a cornerstone from which to begin to see what form an objective system of ethics might take, and from whence we can build. To begin, I think we need to see as clearly as possible what is going on in mathematics and logic that allows us to be able to call them objective and universal. In this we will see that there are two different truth claims being made. One in terms of systematic coherence and consistency and a different kind of truth claim is being made in applications and in modeling in particular. Through an analysis of mathematical modeling I hope to show some parallels between the applied mathematician and the socalled ‘naturalized’ moral philosopher, where the ethicist can be conceived in line with the social scientist. I also consider some of the Wittgensteinian concerns of the moral particularists as they relate to the tension between theory and practice. Next, I will briefly discuss Amy Gutman’s ‘Deliberative Universalism’ as an example of a theorist who has considered and incorporated into her scheme a balance between pure theory and blind practices. In conclusion, I hope to show that if ethics is objective in the same way that mathematics is objective ( as Russ Shafer-Landau claims ), then this implies the use of some similar formal axiomatic framework as well as varying degrees of probabilistic accuracy within real world applications. “For a claim to be objectively true, it must be true independently of what anyone, anywhere, happens to think of it. Mathematical truths are like this,...I believe that moral truths enjoy the same status as the truths I’ve just mentioned: they are objectively true.” (Shafer-Landau, Russ; Whatever Happened to Good and Evil? pg. 11-12) How is it that mathematics must be true independently of what anyone, anywhere, happens to think of it? Mathematics is a highly formalized axiomatic system of definitions, axioms, conjectures, proof or justification, lemmas, theorems, and corollaries. Axioms are basic formal assumptions concerning the objects defined to be in the subject domain. A conjecture is a claim concerning the nature or some relation of the objects in the subject domain. A theorem is a conjecture that has been proven or justified in terms of the definitions and axioms of the system and corollaries are evident and immediate results of a given theorem. The truths presented to us in these theorems are true with respect to the consistency and overall coherence with the other theorems, definitions and axioms. The axioms are the rock solid foundation within such systems. They keep us out

of infinite regress in that all of the justification of any given theorem can be broken down and explicitly explained strictly in terms of the given axioms. The theorems are in some sense just an unpacking of the relationship between the axioms and the definitions. If we change any axiom of a given system, a shockwave is sent throughout the system, potentially, causing radical changes in the character of the derived theorems which ultimately effects the entire structure of the system. One may note that I am taking up a kind of formalist approach towards mathematical truth. “According to formalism, on the other hand , there are no mathematical objects. Mathematics just consists of axioms, definitions and theorems – in other words, formulas. In an extreme view, there are rules by which one derives one formula from another, but the formulas are not about anything; they are just strings of symbols. Of course the formalist knows that mathematical formulas are sometimes applied to physical problems. When a formula is given a physical interpretation, it acquires a meaning, and may be true or false. But this truth or falsity has to do with the particular physical interpretation. As a purely mathematical formula, it has no meaning and no truth value.” ( Davis, Hersh, and Marchisotto, The Mathematical Experience, pg 357) In this project I don’t want to get to much into the details of formalism. I do want to say, though, that I am promoting a somewhat weaker version of formalism than what we see here. I want to allow for truth to be present in purely mathematical concepts and would like to flesh out the difference between this sort of truth and what we see when these concepts are being used to describe something in the real world and thus given a physical interpretation. The kind of truth that we see emerging in conceptual, axiomatic systems, I think, is best explained in terms of the coherence theory of truth. A coherence theory of truth places a high priority on consistency and coherence with respect to the system as a whole. The fact that no one person has any special epistemic access to some realm of absolute truth, suggests that it is through some sort of framework, regardless of how formal, that agents are able to acquire and asses their understanding of the various things in the world. Mathematics is a highly formalized example of just such a framework. The truth within the framework is so in relation to consistency and coherence with the overall conceptual scheme. The justification of the framework can be looked at pragmatically in terms of its ability to provide us with useful tools for describing and understanding actual things in the world. Although, when the framework is being applied to things in the world the matter of how adequate the description is involves a different notion of truth that we will see when I discuss modeling. This coherence theory of truth does a good job of explaining how mathematics and logic can be true independently of what anybody thinks of it. Once the definitions are understood and the axioms have been accepted the truth of the theorems are immediately necessitated. Mathematics and logic under the coherence theory become a self-referential system of knowledge and understanding. That is that all of the ‘mathematical objects’ are true strictly in terms of their coherence and consistency within the set of accepted axioms underlying the given mathematical system. One might want to say sure, but doesn’t mathematics and logic have something to do with actual things in the world? I would say that it is true that many mathematical and logical projects were motivated by things that

were going on in the world (many were/are not), but once they have been formalized they way that they are in a mathematical or logical framework they will remain true as a system independently of the world. What I’m saying is that logic is still true in the absence of language and that arithmetic would still be true in the absence of quantity. This is the fundamental difference between systems like mathematics and the natural sciences, who have their claims staked directly in terms of actual things in the world; mathematics does not change with our understanding of the actual world, which is precisely why Euclidean geometry is still useful today while the Ptolemaic system of astronomy is not. In this point we may have found the strongest analog between projects like mathematics and logic and our project of ethical objectivity. I see a direct link between the motivations, and the overall approach, for mathematical and logical objectivity and ethical objectivity. In logic the motivation was to take the use of language and the basic rational assumptions implicit within its use and to formalize this activity by placing it within an abstract, generalized framework in which it can be articulated. In arithmetic the idea was to formalize our notions of quantity within the same type of framework from which notions of quantity could be articulated; in geometry shape, in algebra space, in calculus motion and curves, etc... In each case the motivation is coming from concepts that are already found within the world, but lead to a coherent system of truth whose truth is independent of the actual world. Truth being independent from the actual world is not a necessary but a sufficient condition for objectivity. All I mean to say is, that it would be much more of a tragedy for the physicist if things like space and time turned up not to be real than it would be for the mathematician. The point here is to say that the project of ethical objectivity is starting with a similar playing field. That is, the world is already operating off of some set of moral intuitions and ethical conceptions the same way language and reason existed prior to logic and trade existed before formal arithmetic, so moral intuitions exist prior to our framing them in some formal theoretical framework. This is an important point to keep in mind, it provides us with some symmetry between the starting point for our project and many others that have indeed achieved an objective status. It is important to note that logic didn’t change the use of language and so it might be that any theory of ethical objectivity shouldn’t completely interfere with our most basic moral intuitions, although, we should certainly leave open the possibility of discerning moral error. This brings to mind what John Rawls called the ‘reflective equilibrium’ between theory and intuition or practice. In Rawls’s contractualist ethics he recognizes the need to balance what is going on theoretically in academia and what is actually going on in the real world with respect to how agents are situated and living in the world. This is going to be a tension in any theory that is staking claims about how the world actually is or should be. This is especially true in an axiomatic system where through consistency and coherence its truths can obtain the status of objective independently of how the world is. It seems weird though, that we could say that ethical truth would be true independently of the world; anymore than one can say that psychological truths would be true independently of the human experience. Social theories seem to emerge, rather, as a result of the particulars of the world and how agents are situated in it. Social theories can be true only in an axiomatic sense, independently of the world, but to remain true to their claims about things in the world, their theories should in some way do the job of helping to articulate what is actually going on in the world. One can see why Russ Shafer-Landau

defined objectivity in terms of truth being independent of what anyone thought about it. This can be achieved axiomatically, but this doesn’t give us the full scope of our claims about the world. This tension can be clearly seen in mathematics in terms of the tensions between the pure and applied mathematician. It is a fact that over ninety percent of the mathematics being done in universities today have nothing to do with how the world actually is and have very little interest in describing things in the world: the various branches of pure mathematics. It is possible, for example, with modular addition, to redefine how addition works on real numbers in a way that is completely counterintuitive to how most people perceive quantity in the world; this new operation on the set of real numbers forms an algebraic structure or ‘group’ which is isomorphic to the algebraic structure of the real numbers under ordinary addition ( in algebra, an isomorphism is a one to one, onto, operation preserving mapping from one structure onto the other and shows a kind of equivalence.) So, with an axiomatic approach one can easily create consistent systems that have nothing to do with anything in the world, but are as equally true as the ones that do. The pure mathematician has no claims staked in terms of how things actually are in the world. The role of the moral theorist stands in much stronger relation to that of the applied mathematician. Through modeling the applied mathematician stakes claims about how things actually operate in the real world. There are some relevant practical notes about the process of mathematical modeling that I would like to point out. Engaging in the job of constructing a model of some process taking place in the real world always begins at the level of being highly idealized; this is to say that the mathematician always begins with the most simple possible case. If we can’t accurately describe the simpler case, then it is unlikely that we will be able to describe the more complicated situations. So, for the mathematician, there is a very practical side to working in the realm of the ideal and abstract for it serves as sort of a guiding light through the murkiness of the real world. It is my strong opinion that it is much easier to study math than to use math for the very reason that circles and squares don’t emerge naturally in the world; neither do numbers, and other means of classification and categorization emerge naturally; this is to say that the things in the world don’t come ready made with numerical or geometric or taxonomical tags attached to them. This point may seem obvious but it seems highly contested when taken to the realm of the ethical. There is much debate centered around the idea of the natural or descriptive fallacy in ethics, for example. This fallacy, initiated by Hume, is essentially a ban on deriving an ought from an is. In Hare’s world agent utilitarianism it is read in terms of deriving prescriptive statements from pure descriptive statements and thus the term descriptive fallacy. It also falls under the heading of the fact value distinction where the idea is that reason provides us with facts about the world but that our moral judgments are the results of our passions and desires. I’m trying to present a picture of mathematical objectivity with as little metaphysical baggage as necessary. I am saying that sure, there is no fact about the world that necessitates the use of a base ten number system in formalizing our understanding of quantity, but the base ten number system is useful as a tool for achieving such an understanding. There is nothing implicit within the descriptive that implies value in general but neither is there anything implicit that implies number or some other more elaborate mathematical model. These ‘projections’, if you will, are still

essential in terms of how we understand or make sense of what it being described. Likewise, I want to say to Hume, fine, there seems to be no fact about the world that directly implies how the world should be or how we should direct our passions in the world, but the way in which we do apply our passions to the world, regardless of the ‘facts’, plays a large significant role in how we do understand the world and especially our moral world. Further, the descriptive element in ethics plays a useful explanatory role and in the absence of such, ones prescriptions risk becoming meaningless abstractions. This is very much how a strict formalist would view pure mathematical theory. Meaning only emerges within the context of the particulars of some physical application in which a mathematical abstraction becomes meaningful. No one seems to have a problem with the fact number distinction in mathematics in particular. It seems to me that the descriptive fallacy in moral theory is just a variation on the tensions that arise between theory and practice. Reason plays a keen role in the descriptive project in ethics, but it also seems crucial to the prescriptive in terms of practical deliberation which is central to an application of any ethical theory. There is a lot to be said also about the role of the particulars of an application in our formulation of an appropriate model. It is by immersing herself in the details of the particulars of the application that the mathematician is able to formulate an accurate model of the given phenomenon. Take for example, the flow of heat through some medium, the simplest case would be that of a one dimensional rod of finite length, L, that is perfectly insulated laterally and is kept at a constant zero degree temperature at the end points. The model turns out to be a simple partial differential equation, u/t = k /x( u/x ) with boundary conditions, u(0,t)=0 and u(L,t)=0 and initial condition u(x,0)=f(x), where u(x,t) is a function that models the distribution of heat through the rod, k is the thermal conductivity of the material of the rod, and f(x) is the initial heat distribution. The solution is easily obtainable by separation of variables and superposition, it is a Fourier sine series u(x,t) = a_n sin(nx / L) e^(kt). This case is highly ideal though, there is no real case in the world where a medium is perfectly insulated for instance. Then, the mathematician must examine how the heat is dissipating from the medium in each particular case. This puts a heat dissipation term into the simple model making it nonhomogeneous and much more complicated to find a solution. Also, it is difficult to keep the endpoints constant. If the boundary conditions are not constant, the solution becomes even more difficult to find. So, by the time the mathematician has modified the simple model to the point it is accurately describing a particular case of heat flow in the real world the model often times becomes too difficult to solve explicitly and thus numerical approximation techniques are needed. The question then becomes a matter of stability and error analysis. The point with this is that even though the mathematical model is objectively accurate and true, in the sense that it gives a mathematical description of the phenomenon, the solution that we end up working with practically is merely an approximation to the model. Through analysis we can objectively bound this error so that we know objective things about the approximated solution but it is still nonetheless an approximation, and thus probabilistic. So even in a mathematical prescription, even though we have a nice general model or theory this doesn’t guarantee the existence of an exact solution or answer to a particular case especially a complicated case. This is what I take to be some of the primary concerns of Margaret Olivia Little in her

defense of moral particularism. The idea that some general normative rule or set of rules will always be enough to capture the complexity of what goes into making a moral decision has been a popular part of most of the dominating trends in moral philosophy since Kant. Little argues against this trend in her defense of moral particularism and describes a particularist as; “Such a theorist makes the rather dramatic claim that moral answers cannot be captured in general formulae. She claims not simply that we should be properly attentive to the relevant details of situations before we can apply any rule or principle to them, but rather that there are no rules or principles, even enormously complicated ones, capable of codifying the moral landscape.” (Margaret Little, “Wittgensteinian Lessons on Moral Particularism” pg 161-162) I think that if such Wittgensteinian intuitions are taken seriously that it can only be a healthy thing for any one who works in some field that deals with models and general theories. This attitude, if taken seriously, forces the theorist to be honest about what their theories and models really supply about the world. Theories and models don’t give you reality they only provide a formal framework in which clarity can be brought to that aspect of reality that you are interested in understanding. Arithmetic does not in itself constitute a mind independent reality, but is an extremely useful tool for articulating and formalizing our understanding of quantity which is seemingly constituted as we normally think by some mind independent reality namely substance. So if the Wittgensteinian intuitions of those like Little are just taken to say that one should be careful in not confusing the model or the theory about substance with substance itself then this intuition can be healthy at least philosophically in that it can help to keep us researchers honest about how much we are getting out of our models. The claim that any theory or general model is useless in understanding the complex landscape of the particulars within the various real world applications is taking particularism farther than it needs to go. Such a position is counter-intuitive and contradictory to how we actually do mathematics and the sciences. There is much contemporary talk about ‘naturalizing’ ethics and about rolling up our sleeves and getting our hands dirty by immersing our selves into applied ethics, where the true complexity of a moral dilemma becomes evident. Anytime we approach any situation in which we are trying to understand something about it, we always are bringing some set of preunderstandings about what we are experiencing. The goal of the formal framework is to make the assumptions explicit; it is to make clear what kind of understanding we are bringing to the project; this further provides a reference in situations where a premise is in need of adjustment. It is not the use of general theories and models, but rather, the researchers attitude towards and understanding of such, where the Wittgensteinian lessons provide great philosophical insight. One person who, I think, has done an excellent job of capturing these sentiments is Amy Gutman in her paper “The Challenge of Multiculturalism in Political Ethics” where she defends what she calls a ‘Deliberative Universalism’. Gutman leaves room for the guidance of general rules or principles but also realizes that not all moral conflict or disagreement can be resolved by such rules and will thus be left up to some process of deliberation. She writes,

“Unlike comprehensive universalism, deliberative universalism explicitly recognizes that some conflicts over social justice cannot now ( or perhaps ever ) be resolved by a comprehensive, universally justifiable set of substantive standards. These conflicts are best addressed and provisionally resolved by actual deliberation, the give and take of argument that is respectful of reasonable differences.” ( Gutman, Amy “The Challenge of Multiculturalism in Political Ethics” pg197) Here Gutman leaves room for substantive principles of morality and justice which she claims will be unreasonable to reject or else up for deliberation. Her position also consists of a set of procedural principles to support the deliberation process which is designed for those conflicts in which reason has come to a standstill. Deliberation is necessary, for example, in those cases in which an equally compelling or equally reasonable claim is made on each parties behalf, but where each claim conflicts with the interests of the other.Gutman provides a realistic account of how a naturalized ethicist would operate in the real world. Gutman does a good job at showing us how the substantive and the procedural can be made to work together. Clarity can be brought to these substantive rules and procedural guidelines precisely by the application of just such a formal framework as what we have seen with the axiomatic approach in logic, mathematics, and many of the sciences. Deliberative universalism recognizes that this will not always provide all of the resolutions to all of the various moral conflicts. The use of the axiomatic framework will only help us formalize our understanding of the substantive rules we are presupposing and to examine their overall consistency and general coherence within our particular conceptual scheme and to recognize a conceptual scheme simply for what it is. Applied to the procedural we will come to have a clearer understanding of what guidelines tend to be most effective under various kinds of circumstances. We will also be provided with a framework that will allow us to compare and evaluate varying procedural schemes. Placing the substantive rules and procedural principles within the project of deliberative universalism, within a formal axiomatic framework, would give the applied ethicist the same grounds for claiming objectivity as does the applied mathematician. We both use the realm of the abstract and ideal as a way of making clear and articulating how we understand our conceptual schemes. In applications we both must immerse ourselves into the details of the particular application and in the complex cases, we must make do with our best approximations to which our models become a probabilistic point of reference. If the system of substantive rules is consistent and coherent with reasonable axioms and the system of procedural guidelines is equally equipped, then given the circumstances and the axioms the necessitated results would be objective in the same sense that mathematical models make objective claims. The probabilistic part is the actual deliberative process and like in modeling the appropriate rules must be molded to reflect the particulars of the given application. So that, even though ones model is objectively true and accurate the solutions that we derive and use are probabilistic approximations. It doesn’t follow from this that such tools are, therefore, all together useless as an explanatory apparatus. It seems that mathematics does have something to offer ethical theory in terms of its status as being objective. The structure of mathematics is the formal set of axiomatic assumptions, definitions, and subsequent logical deductions and in applied mathematics

there comes particular applications in which this formal framework is put to use in a particular way. The ethical theorist can make use of a similar formal framework in order to make their assumptions clear and to articulate the meaning of their moral language. This much can provide the same level of objectivity as that of pure mathematics but in both cases what becomes difficult is how this framework is best to be applied so that we get the most accurate understanding of the phenomenon in question. This is a question central to applied mathematics, science, engineering, and also any sort of ‘naturalized’ ethics. If what Russ Shafer-Landau claims is true, that ethics will turn out to be objective in the same way that mathematics is, then it seems that such an objective system of ethics would make use of some similar axiomatic framework. Its models would likewise give objective descriptions and evaluations of given phenomenon but only approximate, probabilistic solutions instead of the commonly misconceived notion that physical phenomenon possess determinate solutions that cover an entire host of circumstances. With such a picture the concept of objectivity in ethics will cease to be such a mysterious concept and we can start to understand one way in which we can begin to do effective applied ethical research. Bibliography: Davis, Phillip; Hersh, Rueben; Marchisotto, Elena , The Mathematical Experience, Birkhauser, 1981 Gutman, Amy “The Challenge of Multiculturalism in Political Ethics” fr: Philosophy and Public Affairs, Vol. 22, No. 3 Princton University Press 1993 Little, Margaret Olivia, “Wittgensteinian Lessons on Moral Particularism” fr: Slow Cures and Bad Philosophers, ed: Carl Elliot, Durham, Duke University Press 2001 Shafer-Landau, Russ, What Ever Happened to Good and Evil?, Oxford Press 2004 Williams, Bernard, Ethics and the Limits of Philosophy, Harvard University Press 1985