Unspeakable Sin
Back when I was an undergraduate and only just starting out on the mathematical path, I learned a fair amount of vector calculus, topology and manifold theory from a rather entertaining (some say peculiar) man named Mike Alder. Now, anyone who believes that reading pure maths notes cannot make you laugh is invited to peruse Mike's manifolds notes (and, yes, that is an aweful lot of stuff for a one semester undergraduate level course). Sadly, he has not made his algebraic topology notes available online.
But this is all besides the point. Mike sometimes moonlights as a writer for Philosophy Now. In one such article, he considers the case of someone he once knew who refused to accept modus ponens as a valid rule of inference. In Mike's words:
"I was pretty clear therefore that Miss B had done something heinous. Whether it merited a prison term I was not quite clear. Maybe it was something where pointing out the error of her ways and deporting her to Australia would be enough. But it was very bad. Immoral if not illegal"
At any rate, it is a rather entertaining article. Those who do not like reading opinionated stuff when the opinion is not their own are hereby warned. If you have the moxy, then go on to read:
The Compleate Logician or Miss Blakemore's Unspeakeable Sin, Mike Alder
Also, picking up on a thread from logicandlanguage.net, Mike has an article available wherein he considers the distinction between science and philosophy and why many scientists try as hard as possible to distance themselves from philosophy. Again, this article comes with a warning for those sensitive to a bit of criticism:
Newton's Flaming Laser Sword or Why Mathematicians and Scientists don't like Philosophy but do it anyway, Mike Alder
At least he 'fesses up and says "You might, just possibly, have been able to detect a touch of intellectual snobbery in this view of philosophy".
posted by Jon @ 11:06 AM
6 Comments:
Jon --
Regarding Miss Blackmore: There has been work in philosophy and in anthropology on whether and why we should accept deductive inference rules, both in general and Modus Ponens in particular. A place to start is Lewis Carroll's (Charles Dodgson's) 1895 article in Mind, "What the tortoise said to Achilles" (Mind ns, 4 (14): 278--280). I think this paper is available on jstor.
The philosopher Susan Haack took up the question of how you could justify the use of Modus Ponens to someone who doubted it: If you justify it with examples of its successful application, you are using a form of induction, which hardly seems strong enough for a deductive rule. On the other hand, if you use truth table with generic variable-names, then you risk using Modus Ponens itself in the justification. (See: "The justification of deduction", Mind (1976),85: 112--119.)
Within anthropology, evidence has been presented that not all societies use MP. Some critics have disputed this evidence, but my reading of the anthrop literature is that the people saying MP is not universal have a case. Peter Gardenfors quotes a study of the reasoning of Uzbeki horsemen, for example, who will only infer the conclusion from an application of MP in the case where they themselves have personal experience of the premises. In other words, MP is not valid for them if the premises are second-hand reports. I can give you references to this literature if you wish.
Posted by Peter
I should add that I believe the Uzbeki position is perfectly defensible, and thus rational (in the sense in which philosophers use this term), in certain circumstances. Certainly, one can envision computing applications which would require it, and it is not dissimilar to the approach of linear logic. In that case, some MP conclusions would not be able to be drawn because one had exhausted one's current stock of instances of the premises.
It is only our 150 years of thinking about logic in a certain way which makes these ideas seem strange. If Boole, de Morgan, Frege etc had started with linear logic (or Hintikka's Independence-Friendly logic, et al) we would find be writing articles in "Philosophy Now" about the weirdness of standard propositional logic.
Posted by Peter
And now I've read Alder's article on Newton. I think he has the actual Isaac Newton entirely wrong! Alder does say that "Not even Newton was a complete newtonian", but I think this understates the situation considerably.
A more accurate statement would be "Newton was not in any sense of the word a newtonian". Why? Because Newton did not think he was doing modern science (testing hypotheses about the real world by observation). Rather, he had strong (and non-conformist) religious beliefs and thought he was identifying God's laws through reading ancient texts, through alchemy, and through mathematical reasoning. He makes completely untestable and untested claims -- eg, that the motion of the planets could be understood with experiments using pendulums here on Earth. (The assumption of a universal gravitational force is entirely bizarre, if you think about it.)
Alder criticizes philosophers for being out of date in thie article, but his view of Newton is at least 50 years out of date. Pity.
Posted by Peter
Thanks for the comments Peter.
From a philosophical point of view, propositional logic *is* pretty weird. But I am not enough of a philosopher to have anything deep to add. I am, however, reminded of the charming dialogue "They're made of meat" which flips the tables on the whole "computers cannot really think" argument.
The basic postulation of the Newton article, namely that traditional philosophical methods are not suited to the natural world is of course not unreasonable. This is precisely why those subjects broke away and developed their own techniques in the first place! This is not to say that philosophical methods are not without their merits. For instance, there is only so much that one can say about the nature of time by using, say, general relativity or temporal logic. Attempting to understand how people actually perceive time is a whole other kettle of fish.
Posted by Jon
And another thing (I'm beginnning to sound like a London cabbie!) --
I know this is probably not Mike Alder's fault, more likely the magazine editor's: His article on Newton has a drawing of Newton saying "\forall \epsilon > 0, \exists N \in Z+ \ldots". But the epsilon-delta semantics for the differential calculus syntax was only devised in the 19th century, by Cauchy and Weierstrauss, 200 years after Newton.
Posted by Peter
I was one of Mike's PhD students. When I entered uni as an undergraduate, Mike had coauthored a set of lecture notes on linear algebra. Alternating blocks of white and gold pages; if you read the white pages, you had enough to pass the unit; if you read the yellow pages, you, well ...
... and this was years before The Matrix, but after following up the "Can you solve Mu" puzzle to Godel Escher Bach, and working through the material (which in the second volume included the Perceptron Convergence Theorem) I was never quite the same again (At that, I was outclassed - one of my friends went onto implement a neuron, in Object Pascal, from scratch.).
The algebraic topology and robotics notes were being marketed as e-books, though it looks like the vendor has gone bust.
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