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That 17 is prime strikes us as absolutely certain, yet there’s nothing in the physical world we can point to as the source of that certainty. (Seventeen blocks

That 17 is prime strikes us as absolutely certain, yet there’s nothing in the physical world we can point to as the source of that certainty. (Seventeen blocks that can’t be arranged into a rectangle? Give me a break.) In that respect, math seems more like subjective experience than science: you might be wrong about the sky being blue, but you can’t be wrong about your seeing it as blue. Here's where a little philosophy would help; we could save a lot of time by invoking the sensible/intelligible distinction. Mathematical entities are intelligible, not sensible, and their mapping to the external world is epiphenomenal. Seventeen is prime, not because of any empirical evidence, but because it has to be prime under the definition of primality. Math, like chess, is a game with rules.

shasta 16y ago
Did that really help? On my screen, your paragraph is about 10 characters longer than his, and about twice as opaque. > Seventeen is prime, not because of any empirical evidence, but because it has to be prime under the definition of primality. This is rather question begging. 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. There are mainly two reasons why we believe these rules to be consistent: we've used the rules over and over and have never found any contradiction, and we perceive the rules to be about something - the natural numbers. If you perceive the natural numbers the way that I do, then it should seem clear to you that searching for a contradiction in arithmetic is a fool's errand. So are these natural numbers that we perceive "real"? How do we perceive them? My opinion is that philosophy of such matters is a rather hopeless endeavor. I'm sure the philosophers have lots of new definitions and concepts built up to describe the situation, but how much progress can you really hope to make here?
- Groxx 16y ago
  >The real question is why following the rules of arithmetic/primality never seems to lead to contradiction. Because they're built up, not discovered down. With some exceptions. But by and large, this is the precise opposite of any empirical (ie: most) sciences which people are aware of. And, as long as the core axioms are valid, it will be consistent - there's no "belief" in the products, only in the axioms.
  - shasta 16y ago
    Can you elaborate or provide a reference explaining your use of "built up" vs. "discovered down"? I don't understand what you mean, and those terms don't look google-able. > And, as long as the core axioms are valid, it will be consistent Are you using "valid" informally here? Formally, validity is about logical truth under any interpretation of the symbols of a logic. I don't see how that applies, since most statements about the natural numbers do not hold for arbitrary objects. My guess is that you're just making the statement, "we don't have to believe in conclusions, only the premises - the conclusions follow logically." Logical reasoning just provides a way for us to extend our beliefs. If we believe in some axioms, and we believe some conclusions follow logically from those axioms, then we tend to believe in the conclusions as well.
    - Groxx 16y ago
      Math is built from the axioms. From a minimal set, building (provably accurate) complexity. Empirical sciences try to deduce axioms from complex behaviors. From a maximal set, (apparently / hopefully accurate) reductions in complexity. And of course there is a bit of both in each other. Sciences would be essentially valueless if they didn't use those reductions to predict unseen behavior, and there are different sets of mathematical axioms to represent different ways of looking at things / different forms of math. Belief forms the bases of both, true, but once those basic beliefs are accepted, math is consistent. Most sciences are not, as they require belief that this experiment is accurate, as well as all previous ones it is based on, individually, as any may introduce error into the next and all accept that they cannot account for everything as they cannot observe everything. A mathematical proof does account for everything, relying only on the core axioms being accurate.
      - shasta 16y ago
        Yes, that's the story that working mathematicians use day to day, but glosses over the relevant question in this context - why do we believe in one set of axioms instead of another? Do you think it's possible that we'll find a factorization of 17 tomorrow, and so will just need to refine our axioms?
- michael_dorfman 16y ago
  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?

shasta 16y ago

Did that really help? On my screen, your paragraph is about 10 characters longer than his, and about twice as opaque. > Seventeen is prime, not because of any empirical evidence, but because it has to be prime under the definition of primality. This is rather question begging. 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. There are mainly two reasons why we believe these rules to be consistent: we've used the rules over and over and have never found any contradiction, and we perceive the rules to be about something - the natural numbers. If you perceive the natural numbers the way that I do, then it should seem clear to you that searching for a contradiction in arithmetic is a fool's errand. So are these natural numbers that we perceive "real"? How do we perceive them? My opinion is that philosophy of such matters is a rather hopeless endeavor. I'm sure the philosophers have lots of new definitions and concepts built up to describe the situation, but how much progress can you really hope to make here?

Groxx 16y ago
>The real question is why following the rules of arithmetic/primality never seems to lead to contradiction. Because they're built up, not discovered down. With some exceptions. But by and large, this is the precise opposite of any empirical (ie: most) sciences which people are aware of. And, as long as the core axioms are valid, it will be consistent - there's no "belief" in the products, only in the axioms.
- shasta 16y ago
  Can you elaborate or provide a reference explaining your use of "built up" vs. "discovered down"? I don't understand what you mean, and those terms don't look google-able. > And, as long as the core axioms are valid, it will be consistent Are you using "valid" informally here? Formally, validity is about logical truth under any interpretation of the symbols of a logic. I don't see how that applies, since most statements about the natural numbers do not hold for arbitrary objects. My guess is that you're just making the statement, "we don't have to believe in conclusions, only the premises - the conclusions follow logically." Logical reasoning just provides a way for us to extend our beliefs. If we believe in some axioms, and we believe some conclusions follow logically from those axioms, then we tend to believe in the conclusions as well.
  - Groxx 16y ago
    Math is built from the axioms. From a minimal set, building (provably accurate) complexity. Empirical sciences try to deduce axioms from complex behaviors. From a maximal set, (apparently / hopefully accurate) reductions in complexity. And of course there is a bit of both in each other. Sciences would be essentially valueless if they didn't use those reductions to predict unseen behavior, and there are different sets of mathematical axioms to represent different ways of looking at things / different forms of math. Belief forms the bases of both, true, but once those basic beliefs are accepted, math is consistent. Most sciences are not, as they require belief that this experiment is accurate, as well as all previous ones it is based on, individually, as any may introduce error into the next and all accept that they cannot account for everything as they cannot observe everything. A mathematical proof does account for everything, relying only on the core axioms being accurate.
    - shasta 16y ago
      Yes, that's the story that working mathematicians use day to day, but glosses over the relevant question in this context - why do we believe in one set of axioms instead of another? Do you think it's possible that we'll find a factorization of 17 tomorrow, and so will just need to refine our axioms?
michael_dorfman 16y ago
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?