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Did that really help? On my screen, your paragraph is about 10 characters longer than his, and about twice as opaque. It's only opaque if you haven't read Kant

Did that really help? On my screen, your paragraph is about 10 characters longer than his, and about twice as opaque. It's only opaque if you haven't read Kant; if I am going to have to explain Kant to you, it's going to take a hell of a lot more characters than the extra 10. There are plenty of sets of rules that are inconsistent. The real question is why following the rules of arithmetic/primality never seems to lead to contradiction. Why is that the question? I don't know of any mathematicians particularly interested in that question at all, actually. So are these natural numbers that we perceive "real"? How do we perceive them? Not, that is a more interesting question. My answer is already implied by what I wrote above (in Kantian terms). We don't perceive natural numbers with the senses. A number is not a physical thing in the world-- it is an abstraction. We've also invented a series of manipulations we can perform on these abstractions, much like we have invented the rules of chess. And, there are a lot of extremely interesting things that come out of our manipulating the abstractions according the the rules we've set up. However: asking why the number 17 is prime, is like asking why a bishop can only move diagonally. The bishop moves diagonally because that's what a bishop does. Seventeen is prime because that's what prime means. We don't need recourse to the external world of empirical sense perception to answer these questions. Was that less opaque?

shasta 16y ago
(First, apologies for use of "opaque" which was needlessly inflammatory.) I've read "A critique of pure reason" some years ago, but honestly I found much of it to be rather dodgy. Take this: >We don't perceive natural numbers with the senses. A number is not a physical thing in the world-- it is an abstraction. We've also invented a series of manipulations we can perform on these abstractions, much like we have invented the rules of chess. And, there are a lot of extremely interesting things that come out of our manipulating the abstractions according the the rules we've set up. I see this as new words describing the situation, but very little progress. Are these abstractions "real"? How do we perceive them? I agree it's not with our eyes and ears. > However: asking why the number 17 is prime, is like asking why a bishop can only move diagonally. The bishop moves diagonally because that's what a bishop does. "That's just the way it is" isn't an explanation of anything, though. How do we know, once we've proven 17 prime, using these rules, that we will never discover a factorization of 17? How do we know that our reasoning about these abstractions is sound?
- michael_dorfman 16y ago
  Are these abstractions "real"? That depends, of course, on how you define "real". Are the rules of chess "real"? How do we perceive them? I agree it's not with our eyes and ears. And here we're back to Kant. They are intelligible, not sensible. We perceive them with our mind/intellect. (Interestingly, in Buddhist philosophy, there are six senses-- the mind joining the other five, so they push the intelligible/sensible distinction down a level.) "That's just the way it is" isn't an explanation of anything, though. How do we know, once we've proven 17 prime, using these rules, that we will never discover a factorization of 17? How do we know that our reasoning about these abstractions is sound? I'm not saying "That's just the way it is"-- I'm saying "That's the way it has to be because we defined it that way." How do we know that in some particular game of chess, the bishop won't suddenly be allowed to move sideways? Anyway: if you are really interested in the reduction of arithmetic to pure logic, check out Russell and Whitehead's Principia Mathematica-- or better yet, read the graphic novel "Logicomix" which does a wonderful job of covering the domain for non-specialists.

shasta 16y ago

(First, apologies for use of "opaque" which was needlessly inflammatory.) I've read "A critique of pure reason" some years ago, but honestly I found much of it to be rather dodgy. Take this: >We don't perceive natural numbers with the senses. A number is not a physical thing in the world-- it is an abstraction. We've also invented a series of manipulations we can perform on these abstractions, much like we have invented the rules of chess. And, there are a lot of extremely interesting things that come out of our manipulating the abstractions according the the rules we've set up. I see this as new words describing the situation, but very little progress. Are these abstractions "real"? How do we perceive them? I agree it's not with our eyes and ears. > However: asking why the number 17 is prime, is like asking why a bishop can only move diagonally. The bishop moves diagonally because that's what a bishop does. "That's just the way it is" isn't an explanation of anything, though. How do we know, once we've proven 17 prime, using these rules, that we will never discover a factorization of 17? How do we know that our reasoning about these abstractions is sound?

michael_dorfman 16y ago
Are these abstractions "real"? That depends, of course, on how you define "real". Are the rules of chess "real"? How do we perceive them? I agree it's not with our eyes and ears. And here we're back to Kant. They are intelligible, not sensible. We perceive them with our mind/intellect. (Interestingly, in Buddhist philosophy, there are six senses-- the mind joining the other five, so they push the intelligible/sensible distinction down a level.) "That's just the way it is" isn't an explanation of anything, though. How do we know, once we've proven 17 prime, using these rules, that we will never discover a factorization of 17? How do we know that our reasoning about these abstractions is sound? I'm not saying "That's just the way it is"-- I'm saying "That's the way it has to be because we defined it that way." How do we know that in some particular game of chess, the bishop won't suddenly be allowed to move sideways? Anyway: if you are really interested in the reduction of arithmetic to pure logic, check out Russell and Whitehead's Principia Mathematica-- or better yet, read the graphic novel "Logicomix" which does a wonderful job of covering the domain for non-specialists.