"'you can't believe, for instance, that mathematics and poetry are interchangeable!'
"'No,' said Raven." (p. 598)
A mathematics undergraduate once told me that anything could be said in mathematics. I suggested that it might be necessary to invent a lot of new terms and also to formulate a lot of very complicated expressions. He conceded this. Then I replied that, if I was allowed to invent a lot of new terms and to formulate a lot of very complicated expressions, then I would able to say anything in English. He did not have any comeback to this. He failed his degree. Many years later, I recounted this conversation to a guy who was doing post-Doctoral research in mathematics and who later wound up analysing data from the Hubble Space Telescope. He replied that maths has its limits and cannot express everything and he was not surprised that someone who said that failed his degree.
I met a graduate in another subject who dismissed the proof that there is no highest prime number as "goobledygook" merely because he did not understand it at first reading. I did not understand it at first reading but I know my limits as a mathematician. This guy also understood neither logic nor scientific method. Maybe all undergraduates need some kind of introduction to basic reasoning processes?
6 comments:
Kaor, Paul!
I'm sure as heck no mathematician, but I do know a number like 2 has a higher number, 3, and so on infinitely.
I have sometimes thought it would be a good idea if basic education reverted to the system used in the Middle Ages: after learning how to read students began their more advanced education studying grammar, rhetoric, and logic.
I also thought of how Anderson used geometry in "The Three Cornered Wheel" and the use he made of mathematics in Chapter 8 of ENSIGN FLANDRY.
Ad astra! Sean
The argument was about there being no highest *prime* number.
The proof of that isn't totally trivial, but I don't see how anyone can dismiss it as 'goobledygook'.
The proof is a fairly elementary case of proof by assuming the contrary & showing one can derive a contradiction.
http://delphiforfun.org/programs/math_topics/proof_by_contradiction.htm
As that link says this proof is at least as old as Euclid.
A prime number is divisible only by itself and one. Thus, 1, 2, 3, 5, 7, 11, 13 etc are prime whereas 2, 4, 6, 8, 9, 10, 12 etc are products of primes.
Every number is a prime or a product of primes. At any time, there is a highest known prime = p. How do we know that p is not the highest prime? Calculate (factorial p) + 1 = q. (Factorial p = 1 x 2 x 3... up to and including p.)
If q is a prime, them it is a prime greater than p. If q is not a prime, then it is divisible by a prime greater than p because, if you divide q by p, then you get a remainder of 1. Either way, there is a prime greater than p for any value of p. There is an infinity of primes.
Kaor, Jim and Paul!
Yes, but numbers, primes or not, can progress infinitely. Both positive and negative numbers.
Ad astra! Sean
Sean,
But it had to be proved that there was no highest prime.
Paul.
We know that, after every odd number, there has to be another even number and vice versa but the prime numbers are distributed with apparent randomness along the number line. Until a proof was formulated, there was no way of knowing whether the primes were finite or infinite.
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