5 ms·

It’s science if it bites back

33 points by gnosis 16y ago

fragmede 16y ago
> Seventeen blocks that can’t be arranged into a rectangle? Give me a break. Give me a break. Is multiplication anything but shorthand for counting the number of blocks in an M by N rectangle? So then, how is that not an adequate whatever for 17 being prime? It may not be be the traditional math you learned in school alongside long division and multiplication, but it's math all the same.
- ent 16y ago
  Multiplication is a bit more general thing than that. Take multiplication of fractions, for example. You can't express that as the number of discrete blocks in a M by N rectangle. Of course you could generalize M*N to mean "the surface area of the quadrilateral defined by two perpendicular sides with the lengths M and N", which is pretty close to the original. Multiplication with negative or complex numbers will be more difficult to fit into this analogy. edit: some spelling
- Confusion 16y ago
  Is multiplication anything but shorthand for counting the number of blocks in an M by N rectangle? Originally, it probably was (but dive into the history of maths to be sure). After various abstractions and generalizations, it isn't anymore.
  - arethuza 16y ago
    Multiplication in a finite field is quite straightforward, clearly not based on the area of a rectangle and highly relevant to CS.
- RiderOfGiraffes 16y ago
  Suppose you have a number k in mind, and you discover that whenever that number divides a product mn, then your chosen number divides one or the other (or both) of the multiplicands. In shorthand, your number k satisfies k|mn => k|m or k|n Alternative formulation: k|mn and not(k|m) => k|n This is an interesting property to have, and has arisen several times in different contexts and different places. 17 is a number like that, and so are 37, 41, 43, 101, 103 and 257. Such numbers as k are called "prime."
- Avshalom 16y ago
  [][][][][][][][][][][][][][][][][] Bam a 17x1 rectangle.
- shasta 16y ago
  > Is multiplication anything but shorthand for counting the number of blocks in an M by N rectangle? His point is that the notion of primality doesn't have anything to do with the properties of real blocks. Primality tables are not constructed in the laboratory by teams of scientists with giant collections of blocks. Real world objects may have motivated the idea of multiplication and primality, but the idea ultimately does not depend upon any particular real world objects.
michael_dorfman 16y ago
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.
Groxx 16y ago
They seem surprised that mathematicians, who deal very strictly in almost pure logic, ply their skills in the philosophical realms. I highly doubt this has anything to do with math's "mystical tendencies" and more to do with similar skill sets. Also, apparently they haven't taken even moderately high level math courses, which give much more concrete definitions of divisibility. 17 is prime because it matches the definition of "prime". It also doesn't match the definition of "even". What of it? Blue is blue because it's blue, not because it can't be arranged into a rectangle.

33 points by gnosis 16y ago

> Seventeen blocks that can’t be arranged into a rectangle? Give me a break. Give me a break. Is multiplication anything but shorthand for counting the number of blocks in an M by N rectangle? So then, how is that not an adequate whatever for 17 being prime? It may not be be the traditional math you learned in school alongside long division and multiplication, but it's math all the same.

ent 16y ago
Multiplication is a bit more general thing than that. Take multiplication of fractions, for example. You can't express that as the number of discrete blocks in a M by N rectangle. Of course you could generalize M*N to mean "the surface area of the quadrilateral defined by two perpendicular sides with the lengths M and N", which is pretty close to the original. Multiplication with negative or complex numbers will be more difficult to fit into this analogy. edit: some spelling
Confusion 16y ago
Is multiplication anything but shorthand for counting the number of blocks in an M by N rectangle? Originally, it probably was (but dive into the history of maths to be sure). After various abstractions and generalizations, it isn't anymore.
- arethuza 16y ago
  Multiplication in a finite field is quite straightforward, clearly not based on the area of a rectangle and highly relevant to CS.
RiderOfGiraffes 16y ago
Suppose you have a number k in mind, and you discover that whenever that number divides a product mn, then your chosen number divides one or the other (or both) of the multiplicands. In shorthand, your number k satisfies k|mn => k|m or k|n Alternative formulation: k|mn and not(k|m) => k|n This is an interesting property to have, and has arisen several times in different contexts and different places. 17 is a number like that, and so are 37, 41, 43, 101, 103 and 257. Such numbers as k are called "prime."
Avshalom 16y ago
[][][][][][][][][][][][][][][][][] Bam a 17x1 rectangle.
shasta 16y ago
> Is multiplication anything but shorthand for counting the number of blocks in an M by N rectangle? His point is that the notion of primality doesn't have anything to do with the properties of real blocks. Primality tables are not constructed in the laboratory by teams of scientists with giant collections of blocks. Real world objects may have motivated the idea of multiplication and primality, but the idea ultimately does not depend upon any particular real world objects.

michael_dorfman 16y ago

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.

Groxx 16y ago

They seem surprised that mathematicians, who deal very strictly in almost pure logic, ply their skills in the philosophical realms. I highly doubt this has anything to do with math's "mystical tendencies" and more to do with similar skill sets. Also, apparently they haven't taken even moderately high level math courses, which give much more concrete definitions of divisibility. 17 is prime because it matches the definition of "prime". It also doesn't match the definition of "even". What of it? Blue is blue because it's blue, not because it can't be arranged into a rectangle.