Software Musings

What Euclid said to the Tortoise

Posted on: 16/12/2006

I’m reading Brian Leiter’s rather subtly entitled The Future For Philosophy. Note that this is not The Future OF Philosophy but rather the future FOR it. Not that most of the articles seem to bear much relationship to the book’s title. There are several interesting articles therein but the one on which I’ve spent the most time is Philip Petit’s article on Existentialism, Quietism and the Rôle of Philosophy (I’ve dropped the Oxford Comma from the title as it “presents more problems of CrossPondian translation than any other form of punctuation”).

Petit’s article is largely about inhowfar philosphical speculation could or should affect our interaction with the real world. He begins by quoting Kierkegaard saying “In relation to their systems most systematisers are like a man who builds an enormous castle and lives in a shack close by; they do not live in their own enormous systematic buildings.”

There is one claim in Petit’s article that has delighted me and one issue that has really made me think.

The claim relates to the infinite inferential regression in Lewis Carroll’s What the Tortoise said to Achilles. Petit puts the Tortoise at the top of an, admitedly unordered, list of “prominent examples” of “…the most influential—though not necessarily compelling—contributions to philosophy in the last hundred years or so”. The “…or so” allows Carroll’s paper, written in 1895, to be included. I have long been impressed by Carroll’s argument and decided to include it, to shew how even modus ponens is not as clear-cut as one might think, in my children’s mathematics book (see this blog passim). In exploring what is happening these days with the Tortoise I found Takahiro Isashiki’s 1999 article What Can We Learn from Lewis Carroll’s Paradox? which tries to separate cleanly arguments of language and metalanguage. My problem, however, was simpler. In the first half of the 20th century school children would have been familiar with Euclid’s axioms and would have known where Carroll’s “Things that are equal to the same thing are equal to each other” comes from. My, very second-hand, copy of Euclid’s Elements is subtitled “Preliminary and Junior Courses, 1906/1907 School Year”. I’m not sure what age that addressed but I assume it was quite young. Today I don’t know any young people who have heard of Euclid, let alone his axioms. So, I had to make the call of whether it was worth explaining The Elements so that I could introduce my readers to The Tortoise so that they could understand that even modus ponens could be doubted so that…. If only Carroll had chosen a more every-day example of a syllogism for the Tortoise to doubt, but I assume that, as a Mathematics Lecturer, this one was second nature to him and potential readers.

Carroll’s article also raises the question of whether a refereed journal (Mind) would today accept an article only a page and a half long written in the form of a dialogue between a Tortoise and a mythical Hero.

Above I mentioned that, in addition to the main thread of Petit’s paper, there were two threads that interested me. The other was when he distinguishes philosophical questions from, say, scientific ones:

My own view is that philosophy deals with questions on which we are already committed, whether we like it or not, and that this existing committment, controversial as it often is, gives those questions the particular interest they have for philosophers.

This obviously begs the question of where that innate committment arises. I, for example, believe firmly that the questions of philosophy should address what can be known rather than what we can know. The psychological turn that philosophy has taken over the past half century seems to me to be a retrograde step. But where did I get that committment?

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3 Responses to "What Euclid said to the Tortoise"

‘..what can be known rather than what we can know.’ I’m afraid I can’t put a gnat’s leg between those two. If what is meant is ‘what we can know rather than what there is to know’ then it would seem to turn on a question of practicality. However, a philosophy of what we could possibly know seems perfectly valid and could for all we know prove equally fruitful in dealing with questions on which we are already committed. I know where you and Mr. Petit did not get your committment from and that’s from a dictionary.:-)

Well, I’m blowed. I stand corrected v.a.v. committment. I could go back and edit the blog entry, thereby rendering your comment, David, meaningless. But I will take it on the chin like a man. It was I who erred. Not Mr Petit. I misquoted him as well. Now, one could wonder why “committed” has two ts and “commitment” only one…

When I wrote that throw-away line at the end of the blog “…what can be known rather than what we can know…” I was not intending to be subtle but comments, both written and verbal, have indicated that I was.

Without attempting to open up the theory of knowledge, I was trying to distinguish between the work in the first half of the 20th century that seemed to be aimed at the limits of what can be known and the later work on what we could know.

For example, a complete set of axioms for a system containing arithmetic cannot be known. Your meaning/understanding of the word “elephant” cannot be known. These are not physiological limitations of my understanding; they are just unknowable.

Then we have intermediate things like the truth of “At this very point where I am typing this in what is now called Ottawa, at 11:23 AM on the morning of what is now called 15th January 1431 a native American was eating a piece of Moose”. As this cannot be determined, whether it has a truth value or not can obviously be debated.

Then we can come forward to things which can, in principle, be known but which we are (as yet) unable to know. 200 years ago, in the culture of the times, slavery was morally acceptable to many people, including philosophers and divines. In the culture of our times it is unacceptable. Is this a reflexion of an advancing society or will our abhorrence of slavery be revisited in the future and found to be morally wrong in the context of a different society’s mores?

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Disclaimer

The author of this blog used to be an employee of Nortel. Even when he worked for Nortel the views expressed in the blog did not represent the views of Nortel. Now that he has left, the chances are even smaller that his views match those of Nortel.
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