Bias and Naive Philosophy of Mathematics

Based on another long discussion about rationality and the goals of philosophy.

Claim:

  1. Everyone is biased therefore a philosophical program to achieve agreement is nonsensical because the biases will never be overcome on philosophical questions.
  2. Mathematics has achieved greater agreement than philosophy because it has answered basic questions, such as 2+2=4, while philosophy has failed to answer basic questions of any kind - witness the continued argument about philosophical questions.

Against:

1) This is complete misreading of what actual mathematicians do. Mathematicians do not go around arguing about basic questions, 2+2=4, because the actual business of mathematics has moved onto a higher/more abstract level and turned 2+2=4 into an agreed upon assumption.

Similarly for philosophy there is a much larger sphere of agreement about basic facts than there seems. In order for philosophy to occur there has to be agreement about what questions are in and out of bounds. So a question about whether archival grade paper lasts longer than normal pulp paper is not considered a philosophical question. Philosophy defines itself into a pretty narrow and common set of areas: epistemology, metaphysics, ethics, logic, aesthetics, philosophy of mind, etc. These are basic behavioral agreements, akin to the behavioral agreement among mathematicians that says questioning 2+2=4 is not a currently debatable/live mathematical question.

2) In order for us to recognize disagreement there has to be so much agreement about so many things that disagreement is almost nonsensical. There are two strands of philosophical argument I know of that defend this point of view.

First, Donald Davidson’s work in “On the Very Idea of a Conceptual Schema.” When I read this paper in my class on relativism I thought it was one of the best philosophy papers I ever read. I still feel that way. Davidson writes:

What matters is this: if all we know is what sentence a speaker holds true, and we cannot assume that his language is our own, then we cannot take even a first step toward interpretation without knowing or assuming a great deal about the speaker’s beliefs. Since knowledge of beliefs comes only with the ability to interpret words, the only possibility at the start is to assume general agreement on beliefs.

Second, phenomenology and Wittgenstein both propose some set of beliefs/knowledge that is widely shared and used in conversation. Phenomenologists call it a lifeworld, Wittgenstein seems to be getting at the same idea when he uses the phrase “forms of life.” From Philosophical Investigations:

241) It is what human beings say that is true and false; and they agree in the language they use. That is not agreement in opinions but in form of life.

3) Disagreement can mean many different things: it could be a mistake, it could be a failed communication, it could be bias, etc. Claiming that philosophical communication or any communication cannot reach agreement because of universal bias ascribes a single cause to a multifaceted problem.

4) Finally there is a pragmatic objection to the overwhelming bias argument. And it comes in two forms. First, so what? Does accepting universal bias really mean that we must abandon any behavior that moves or tries to move us closer to a consensus. Robin Hanson summarizes this nicely when he says “that although it hard to conclusively say anything, you shouldn’t just throw your hands up and say ‘nobody should know anything.’”

Second is the argument over ‘live options’, a phrase used by William James in his discussion of pragmatism. In The Will to Believe he writes:

A living option is one in which both hypotheses are live ones. If I say to you: ‘Be a theosophist or be a Mohammedan,’ it is probably a dead option, because for you neither hypothesis is likely to be alive. But if I say: ‘Be an agnostic or be Christian,’ it is otherwise: trained as you are, each hypothesis makes some appeal, however small, to your belief.

In philosophy there are plenty of questions that present live options to the discussants. In fact these are just the questions that seem to cause philosophy so many problems. Real disagreement is only possible when we have agreed that the question is important.

5) My final point about the differences between mathematics and philosophy is that 2+2=4 and “justice is fairness” or “rationality is a good in itself” are statements of radically different levels of discourse. They may appear to be similar because of their brevity but that compactness hides an entire history of argumentation. Statements about justice or rationality presuppose an entire lifetime of experience with a language in order to even begin to understand what the words might mean. By contrast 2+2=4 is equivalent to the baby talk of mathematics.

We mistake the certainty of mathematics presented in textbooks as something that is eternal and unchanging. But this is historically invalid. Mathematics has changed over time as much as any language. Symbolisms, problems, methods, and more have altered in the past and will continue altering into the future. Just look at the impact of computers on mathematics.

The certainty of mathematics is just an impression of laymen. This certainty appears strong because math depends upon a symbolic language that people learn to wildly different degrees. Philosophy, by contrast, looks simple because it uses the language we use everyday, but it is really as complex and as certain/uncertain as mathematics.

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