Mathematicians and Political Scientists use similar structures in argument presentation, but the different goals of the two disciplines requires the use respectively deductive and inductive logic while the objective versus subjective ideological context of the disciplines results in different understandings of what we consider to be data and how we know it. I first define two important terms and then outline the similarities in the basic models of argument making in the two disciplines. Following this I discuss the different kinds of logic used in the two disciplines and how each uses a logical format that is determined by the purpose of argument making in the respective discipline. Finally I argue that the differences in objectivity versus subjectivity determines the kind of data we can use and the degree to which that data is certain or uncertain. I conclude by arguing that although the disciplines use different logic and have different views of what can be construed as data, we can use techniques of argument making in mathematics to critique and revise arguments in politics.

. In order to begin to unravel the process of argument making we must first define two terms essential to our discussion: deductive and inductive logic. Deductive logic involves reaching a conclusion by reasoning it out, tracing the origins of a statement through to its outcome in a step-by-step, sequential manner with each step dependent on the validity of the prior. Inductive reasoning, in contrast, involves inferring a conjecture and surmising a conclusion from circumstances, evidence and premises. Understanding these two definitions will help us now to take apart argument structure, logical progression, and data.

Although seemingly quite distinct in topic and material discussed, the structuring and revision of arguments in the two disciplines of mathematics and political science is strikingly similar. Mathematicians seek to "prove" something by citing a string of axioms, lemmas, and prior theorems that help to get from what they postulate as their given to what they what to prove via the use of deductive argumentation. In political science at UPS there is in use a similar, though more subjective model of data, warrants, and claims put forth by logician George Toulmin. Under his structure of argument making, data is anything that the audience will give an author. The test of this political equivalent of an axiom is not whether it is the simplest, most reduced statement that can be made as it is in mathematics. Rather the test of data in politics is whether the audience will accept an author’s data as a reasonable statement of truth and is typically either historical or statistical in nature. Toulmin’s method entails that from past observations political scientists work towards an inductive claim, the political equivalent of a "prove this" statement. This claim will always be an inductive statement that they want to end up making about the future, one that is likely but unproven. The logical path that gets from data to claim, called the warrant, is implicit in the axiomatic parallel but must be stated explicitly in politics. Mathematicians proceed from one step to the next through implicit use of deductive logic in argument structure; if the prior step cannot logically support the next, then that step cannot be taken. In politics, though, since authors are trying to induce their way to a claim they need to have a justification for making that induction, and this justification is the warrant. They may make the argument that the claim is true because there are signs or indicators that it might be true, there is a causal effect, generalization of past data indicates likelihood, or an analogy between a known problem/outcome and the question is appropriate. Authors then need to add qualifiers to the argument that can account for the possibility of other outcomes given the uncertainty of the claim, a policy that is anathema to the deductive structure of mathematics. Having laid out the basic similarities of structure in our arguments, we can begin to examine the differences in the types of logic and data in the two disciplines.

In mathematics, the problems defined are the so-called thought experiments, the un-testable mental exercises that Lakatos talks about in his Proofs and Refutations. Seldom can mathematicians carry out mathematical experiments in the real world because potential uncertainty in accuracy and precision means that they may or may not be able to prove what they want to prove. Because they are so hard to test, such mathematical arguments can seldom if ever be induced or experimented on, and so mathematicians are left to examine mathematical argumentation using deductive logic. The attempts to show for sure that the universe is hyperbolic or that parallel lines exist are examples of this problem of accuracy in induction, the result of the vastness of space and limited technology. A diminished capacity for inductive, experimental and empirical evidence presents the need for alternatively deductive logic. Thus when mathematicians seek a problem to answer it is invariably going to be a theoretical one, not in this sense that it is a possible problem but that it exists solely in the mind.

Problems of empiricism aside, deductive reasoning is also necessary in mathematics for philosophical reasons. If mathematicians take an inductive approach to such thought experiments, then they run into all the problems of subjectivity, monster barring and cumbersome counterexamples encountered by the students in Lakatos, and they end up with an incomplete, imprecise proof. Induction by definition leads to speculation about potentialities and since speculation can be reasonably confident but it cannot be certain, inductive reasoning leaving the wonderer in the realm of the subjective. In their realization that they are working with a naïve conjecture with tenets based on their assumptions, Lakatos’ students realize that if a mathematical argument is subjective, then they are working in the middle of the proof instead of the beginning. As the teacher says, "there are no such things as inductive conjectures." Instead mathematicians must refine conjectures until they become deductive, and if they are forced to induce something then they are not starting far enough back in their axiomatic system when they define the problem. The problem for mathematics becomes finding exactly what data to use in deducing that something is true or false, or proving what can or cannot be known. The rules of math state that the problem being asked must be one that is objective and not subject to interpretation, and thus predictive, inductive logic is problematic.

Argument making in politics is quite different from the mathematical argument that seeks knowledge for knowledge’s sake. Political science arguments talk about practical applications towards uncertain future outcomes, and by the fact of that uncertainty such arguments are inherently contestable and inductive. Political scientists ask not what can they know, but what will happen, and simple deduction about past specifics won’t necessarily tell us anything about the future. Instead of looking back to prior data as a way of defining what can be known as in mathematics, arguments in political science look back to contemplate the future. As Diane Schmidt notes in her manual Writing in Political Science, "Inference and inductive reasoning provide the mechanism by which political scientists use data to increase the body of political knowledge." Deductive reasoning is less useful to political science because facts generated by deductive reasoning are not ends in themselves as they are in math; thought experiments along political lines won’t do us much good because they only tell us what is the status quo, not what might happen. Indeed political scientists have a name for such deductive arguments: History. For political science deductive arguments are simply the background for talking about the future, a stepping-stone to forming inductive reasoning about the future. For example, as a scholar I can deduce that Bill Clinton’s healthcare reforms failed because they lacked support in the business community, but this doesn’t tell me anything especially useful in politics. I cannot deduce that business will be a problem in the future because the future isn’t written and other situations may not be met with similar resistance. If however, I can induce that the failure of Clinton’s health care to business interests indicates that future policy with similar opposition in the business community may also fail, then I am getting somewhere. My argument would then be controversial because it is not certain and it would be useful because it provides a theoretical possibility that can be tested against future outcomes. Politics exists not in the theoretical world of Lakatos’ thought experiment but the real world where political scientists need to know about something and predict what might happen. The type of data available in politics further pushes the scholar towards induction and away from deduction, while just the opposite happens in mathematics.

To begin making an argument, an author has to decide where to begin. This may sound simple, but the job at hand is anything but. The question of defining what can be known, what the data is, depends on different factors in the two disciplines and those differences lead us into two very different types of questions. While a mathematician begins his argument by sharpening his argumentative pencil through placing himself along a continuum ranging from conservative dogmatic to liberal concept stretcher, politics adds another continuum of ideology and values, complicating matters and making where political science arguments start dependent on a relative political scale.

It is somewhat ironic that in Lakatos’ Proofs And Refutations it is only at the end that the students finally figure out how to begin. They finally get to the heart of the issue of what it means to know something by discussing what it means to define knowledge in mathematics, the idea that data definition is a compromise between dogmatism and inclusiveness. As Kappa puts it,

In other words, how detailed mathematicians want to be in defining what they know and where they can begin is dependent on their intellectual bias. If they are too dogmatic, too specific or formal, then they eliminate all the data they have to work with. If they stretch what they are willing to call data too much, though, then they have included too much to be precise. That said, though, through refinement and argument mathematicians can come to some understanding of what a rational compromise between dogmatism and concept stretching might look like. Such is the nature of the compromise the students are trying to find. By argument and debate mathematicians may reach agreement on the objective nature of a problem statement. This is the premise of Lakatos’ work, arguing how to define what can be known. Once mathematicians reach an understanding of what they can know they can begin to posit what they might deduce.

Defining what one can know in politics, though, is not nearly so precise. Beliefs and values, unarguable and all more or less equally valid, begin to color the debate over what one can know and make a precise definition of data much more difficult. Define the process of rationally defining the context of a problem statement, outlined in the prior paragraph, as a scale along a line. Politics involves value-based assessments of results and information that come out of statements deduced through the compromise above. It is then necessary then have to add a second dimension, that of the liberal/conservative ideological spectrum, to account for this. Such a model is imprecise but it shows that there are now four areas of the continuum that might offer an opinion of what perspective data might be considered from: conservative dogmatic, liberal dogmatic, conservative concept stretcher and liberal concept stretcher. This moves the model from placing the definition of data from a linear continuum to a plane space. There becomes then a much greater area of the continuum from which to begin arguing about the definition of the data.

Unlike the dogmatist/concept stretcher debate, though, a rationalization of some middle ground between the ideologies is often not possible because no value system is necessarily better than the other and it is in their nature to conflict rather than seek compromise. Writing about beliefs and values, Schmidt notes that, "they cannot be disproved and are not subject to argumentation." Ideological position will delineate what a reader is willing to accept as data just as much as their position as a dogmatist or a concept stretcher. Depending on what an individual’s values are, they will start out at a given place in the political spectrum, likely be unwilling to compromise, and will have their arguments colored by those beliefs. Data can then be taken to mean very different things depending on ideological position. Thus because of ideological differences the battle in politics becomes not how to narrow and define the problem being asked but rather what is perceived as the problem in the first place. Conservatives will say one thing, liberals another, and the problems of one will likely be the praises of the other.

An illustrative policy case illustrates the rather abstract model we have defined here. A political scientist can deduce that welfare roles are declining and while a conservative may induce that there will be less expenditures and sing praises, a liberal will bemoan the people who may suffer poverty when the system fails to assist them. Unlike mathematics, in political science there is no discernable definition of what can be known, only a set of opinions carefully shaped and constantly changing. This then carries over into primary research where political scientists are asking questions that will be answered depending on the biases of the person being questioned. It is not only the actors involved in solving a political problem that face problematic definitions of data but also the authors who have to write about those decisions.

Despite the differences in the two disciplines in the logic used and the way that data is interpreted and determined, it is possible to use one to shine light on the failures and flaws of the other. In political science every author has an argument he believes in, a dedication that can lead him down some slippery paths of ignoring data and presenting less than the full story. Lakatos’ discussion of adjusting mathematical reasoning provides some very useful tools for debunking and reforming such arguments in his discussion of the failings of monster-barring and the benefits of lemma incorporation. Lakatos’ deductive proof analysis helps adjust inductive case studies in politics.

Lakatos defines monster-barring as a redefinition of what we will accept as the data/examples we are using so that we can remove a problem posed by a counterexample which contradicts the argument we are making. The problem with this method then is that we can never be sure we have eliminated all the monsters. Lemma incorporation is the alternate logical solution that resolves the problem of the counterexample without digressing into the infinite regress of monster barring. In lemma incorporation one redefine the tenants of the argument that were contradicted by the counter example rather than redefining what is considered valid data. We limit our argument rather than our data.

In politics I have too often seen great arguments ruined by an author trying to do too much with their argument, trying to make too big of a splash. In doing so they either understate the situation and thus fail to justify their case by failing to consider rather obvious counterexamples, or they overstate the applicability of their case in the face of counterexamples. Two texts that discuss the theory behind public policy making in the United States serve to illustrate these cases, and the contradictions between the two proffered arguments lead into using Lakatos. In Agendas and Instability in American Politics Frank Baumgartner and Bryan Jones argue that politics and policy in America is malleable and that government policy can be changed when the right combination of circumstances arise. As evidence of this they cite the numerous successes in changing public policy on the environment and health related issues during the 1960’s and 70’s. They refer to their model as punctuated equilibrium, arguing that normally stable politics can be fundamentally shifted given the right circumstances. In Government’s End, Johnathan Rauch says just the opposite is true and that the government has become mired in bureaucracy and interest group politics that render it unable to operate in any significant manner. He titles this increasingly degenerative condition Demosclerosis. Each argument thus stands opposed to the other and the reader is left wondering who is right. The answer is that neither is. Both have failed to make their case solid by ignoring the realities observed by the other. An application of Lakatos shows why this is so and how to fix the flaws in both.

By employing Lakatos’ arguments about monster baring and lemma-incorporation I have in my politics course argued that both authors need to lemma-incorporate to save their arguments in the face of the evidence offered by the other. It would be tempting for Baumgartner and Jones to say that America needs to wait and see if Demo sclerosis can be changed by a punctuation of the equilibrium and thus render Demo sclerosis a monster not subject to their argument. Rauch, for his part, would argue that the successes in changing social/environmental policy is relatively irrelevant given the bigger problem of government failings he discusses. He would call Baumgartner and Jones small fish in a big pond and their arguments irrelevant, a monster to be barred. In fact though, these two cases are so diametrically opposed that the existence of counter examples weakens the impact of the arguments and their credibility. Instead the authors need to lemma incorporate and reduce the extent of the applicability of their cases, admitting that their cases aren’t the catch all that they want them to be and reducing the significance of their arguments. Baumgartner and Jones need to exclude reforming interest group politics from the problem that they are trying to talk about and limit themselves to discussing social policy, while Rausch needs to admit that there are cases of social policy that can be addressed in spite of a broader problem of government efficacy. It is then possible to resynthesize the thus limited arguments into a broader understanding of policy. Once we lemma incorporate we can argue successfully that there are some areas of policy that can perhaps be changed but also there are some areas of government that can’t be touched. Such an argument may seem wishy-washy and ambiguous but it is at least a clear understanding of the nature of policy making rather than an overblown claim that cannot be fully supported.

I have argued in this paper for a deductive structure in mathematical argument making based in a tangible definition of what can be considered truth and for an inductive structure in political science arguments based on value-based data. The two disciplines do, however, use a similar format for laying out their respective arguments, and the critical assessment of proof analysis used in mathematics can help inform political arguments. Looking at deciding which form of argument I favor, I am biased towards my political science background. Personal interest aside, though, I find rhetorical argumentation more valuable because it allows us to talk about the future and about change. In mathematics we are interested in an academic pursuit, an understanding of what we can know. Knowledge itself is the end of the kind of mathematical arguments we have discussed here. Political Science, though less certain about the claim being made and about the degree of confidence we can have about what we can know, is more practical. Knowledge it political science is merely a means to an ends of talking about the bigger problem of the future and how to achieve what we want to achieve, or at least knowing what we are likely to be facing in an uncertain world.

As someone who is deeply interested in human social dynamics, interaction, value systems, and religion, I thus find the later style of argument making more useful. Humans, by nature of their often irrational nature, cannot be reduced to certainties necessary for deductive reasoning, and so a mathematical style of argument making about what we can know about human nature is somewhat problematic. More importantly, though, the knowledge as its own virtue element of mathematics is fairly useless when you need to apply your knowledge to uncertain outcomes when dealing with social and political conditions. The very uncertainly and ability to think that separates humans from machines makes humans unpredictable. To the extent that one wants to work in the world and develop a good society, one needs to be able to deduce what has happened and induce what will happen. The deductive approach in mathematics can then get one only so far. However that same deductive approach can be used to go back and critique an inductive approach, asking if the steps we are inducing discuss as much of reality as they can. In the end I would summarize my preferred argument system as one that uses an inductive format of argument to talk about the future and which is based on a deductive analysis of history and informed by criticisms and analysis drawn from deductive logic.

Bibliography

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3. Brockenride, Wayne and Douglas Ehninger. "Toulmin on Argument: An Interpretation and an Application." Quarterly Journal of Speech 1 (February 1960) 44-51
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5. Lakatos, Imre. Proofs and Refutations. Cambridge: Cambridge University Press, 1976.
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