This morning, I had a look at a recent mathematics paper that I am a coauthor of. I was struck by how complex it is. The reasoning in a mathematics paper is extremely elaborate and complex. In philosophy, we tend to think that an argument with, say, twenty steps is very elaborate. But here the proof involves eleven lemmas, each of which has a proof consisting of several, and at times quite a large number of, steps, many of which are quite elaborate. I can see how one can look at a philosophy paper and a mathematics paper, and think: "The mathematics paper, that's really serious intellectual work. The simplicity of even the most complex philosophical arguments, with the exception of ones in philosophical logic, shows the lack of intellectual seriousness of the philosophical enterprise." I think it is not uncommon for scientists and mathematicians to have this attitude towards philosophy.
I think this attitude is mistaken. Anecdotally, writing good mathematics papers is not harder for me than writing good philosophy papers. Writing a mathematics paper takes me significantly longer than writing a philosophy paper. There is a lot more detail. But how long it takes to write a paper is not a good measure of intellectual difficulty or seriousness. Typically (though not always—I think the paper I was looking at is a counterexample) the main difficulty is coming up with the basic idea for the paper. The difficulty in coming up with the basic idea for a good mathematics paper is not very different from that of coming up with the basic idea for a philosophy paper. In both cases, one may spend years thinking about a problem, trying out solutions that fail, and finally the idea may just come—or it may be a series of progressive refinements. Once the idea comes, wrapping it up can be challenging, and in the mathematics case it may involve more tedium (or not—the tedium is different, in the one case there is tedium in getting all the details of the proof right, while in the other case there is a tedium in relating one's result to a vast literature). Of course, one might find that the details aren't as simple as they seemed once one works through them—but this can happen equally in a mathematics and a philosophy case.
Moreover, even in the mathematics case, the length and complexity of a proof is not the mark of intellectual quality. If one could find an elegant, quick proof—that would be all the more appreciated by the community.
5 comments:
In my opinion, the though is that philosophy isn't serious intellectual work because even an extremely complex philosophy paper is nothing more than a rationalization of a personal opinion about some issue that can't really be settled one way or the other. In other words, the thought is that philosophers are just playing around, whereas the mathematicians and physicists are really getting something done. I don't think it's necessarily about the complexity of the work or the amount of toil that goes into it.
Alex,
Interesting thought. I've often had a different anecdotal impression of the views of mathematicians and scientists toward philosophy. I agree that they tend to be dismissive of it, but I've never had the impression they are dismissive because they view it as lightweightish work. Rather, it seems like they tend to think it lacks rigor altogether. A quick illustration, which is fairly representative of what I've seen: A friend of mine (a guy who does primarily phil of mind) was talking to a casual acquaintance of his who was a scientist of some sort. When he told the scientist he did philosophy, the scientist made a sarcastic joke about how it must be nice to sit around all day just sharing your feelings. When my friend pointed out that philosophers give rigorous arguments, albeit perhaps ones that the scientist might think were lightweightish, the scientist just laughed and treated the comment as itself a joke.
(It strikes me as odd that these views can persist even in universities with respectable philosophy departments, where it would be relatively easy for mathematicians and scientists to check out what the philosophers were doing and be disabused of their strange stereotypes. I don't know... maybe my experience is just skewed in some way and mathematicians and scientists are more enlightened about these sorts of things than I've noticed....)
Alex,
This is really interesting for me!
Three long philosophical notes, if I may.
First, too short and clear argument can seem circular (to the opponent). I remember you wrote about this in your papers (discussing your and Dr Gale's new cosmological argument, I guess).
Second, too long argument is an epistemological problem. At least sometimes we are not able to think the whole argument simultaneously. In every moment we think only a part of the whole. E.g., P
is entailed by Q, R, S, T and U. Suppose we (in a given moment) conceive and think entailment of P by T and U. But in this moment we do not conceive and think entailment of T by Q, R and S - so we do not know T, and so neither P.
If we, in every moment, conceive and think only a part of the given whole, then our knowledge of P seems to be threatened. At least, it seems that it is not a CLEAR AND DISTINCT knowledge.
Few years ago, I inquired into Descartes, and I found that he proposes three attempts to solve the problem.
(A) In his Regulae, Descartes has some rules which should enable to understand the whole proof simultaneously. The rules seem be banal and too abstractly formulated, but, as I read somewhere else, Descartes treats some of them more concretely in his scientific texts (Geometry, Dioptrics, Meteors, and the physical part of Principles of Philosophy).
(B) Just use written notes. But the sceptic says: maybe your notes are unreliable; maybe your interpretation of the notes is incorrect. Further, as Bill Vallicella wrote on his blog, "I can write down the steps in an argument, and in this way 'store' the argument. But in order to generate knowledge of the argument's conclusion, I have to read through the steps and retain in short-term memory the premises and subconclusions. So I must rely on my memory. But how do I know that my memory can be trusted beyond the present moment?"
(C) Prove the existence of God and also that falsity of our remembrances is, at least under relevant conditions, incompatible with the existence of God, who supplies our tendency to take them as veracious. My comment is that Descartes' proof of God's existence is problematic, and that the theistic proof of reliability of our remembrances would have to be understood simultaneously.
(By the way, I have in my e-archive this note:
Alex Pruss, discussing with John DePoe at FQI blog, seems to embrace (C) (and also another type of solution); according to Alex, there are two ways out: "prove the existence of God from self-evident propositions, and argue from self-evident propositions that he is unlikely to allow total deception. The second is to start with many of our ethical obligations as primary; note that it is immoral to question some of them; and then note that these obligations entail the existence of other minds, etc. (If I owe things to others, others exist.)")
Third, too long probabilistic arguments for complex conclusions sometimes have low final probability. Consider Plantinga's principle of dwindling probabilities in his critique of historical apologetical arguments for Christianity. The main replies to Plantinga were the following. (1) The probability that every argument for the given conclusion fails can be low (Jason Colwell). (2) Even if the given complex conclusion is less probable than its proper propositional parts, it can be still be probable, whether on the same or on some enhanced body of evidence (R. Swinburne, the McGrews). Finally, (3) there have been detailed probabilistic arguments for many complex conclusions, including Christianity, yielding results significantly different from Plantinga's cursory estimates.
Plantinga's PDP has an analogy in technology and management: the more complex a widget, the greater the probability that it fails; + success requires a conjunction, failure only requires a disjunction (cf. Chris Swoyer in his Critical Rreasoning, ch. 16.6, available at http://www.ou.edu/ouphil/faculty/chris/crmscreen.pdf ). Yet, aren't there many many grievously intricate widgets (e.g. netbooks) which run properly (at least for few years)? Of course, there are.
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