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> The ancient Greeks wanted to believe that the universe could be described in its entirety using only whole numbers and the ratios between them — fractions, or

> The ancient Greeks wanted to believe that the universe could be described in its entirety using only whole numbers and the ratios between them — fractions, or what we now call rational numbers. But this aspiration was undermined when they considered a square with sides of length 1, only to find that the length of its diagonal couldn’t possibly be written as a fraction. Let me try it! If I make a square with sides 1 inch long, then measure the diagonal with a tape measure I get... 1 and 6/16ths! Only whole numbers and the ratios between them involved there, so I guess that's all the Greeks needed after all.

deleted 2y ago
[deleted]
xigoi 2y ago
You can’t do math by measuring real-world objects.
- autoexec 2y ago
  > You can’t do math by measuring real-world objects. Don't we all start out doing math by counting/measuring real-world objects? It was all "Sally has x apples" and rulers when I started school but some kids do math using other tools like cubes and cuisenaire rods which are also used to do math through measuring/counting real world objects.
  - xigoi 2y ago
    Mathematics is a tool that can be used to describe the real world, and mathematical discoveries are often motivated by practital uses. However, mathematical structures are not directly influenced by the real world; they can just model it.
    - autoexec 2y ago
      > Mathematics is a tool that can be used to describe the real world, And that's what the article says the Greeks believed they could do using whole numbers and fractions. In the case of a real world square with sides of length 1 it seems that you can get away with describing the length of its diagonal in those terms which made it seem odd that it was what caused them to abandon their belief/aspiration. Maybe the author just described their dilemma poorly/strangely. I'm not at all suggesting that math has to be limited to the real world or that irrational numbers don't have their place. We're certainly better off with them.
nullc 2y ago
So then you notice that if you measure more precisely you get a different fraction... and even more precisely another different fraction. It's natural to ask what ratio you'd get if you measured with endless precision. In some cases there is such an answer and you can reason it out without conducting a series of increasingly precise measurements (though you could also find it with a sufficiently precise measurement), but in other cases there is no such answer. Isn't that interesting?

deleted 2y ago

[deleted]

xigoi 2y ago

You can’t do math by measuring real-world objects.

autoexec 2y ago
> You can’t do math by measuring real-world objects. Don't we all start out doing math by counting/measuring real-world objects? It was all "Sally has x apples" and rulers when I started school but some kids do math using other tools like cubes and cuisenaire rods which are also used to do math through measuring/counting real world objects.
- xigoi 2y ago
  Mathematics is a tool that can be used to describe the real world, and mathematical discoveries are often motivated by practital uses. However, mathematical structures are not directly influenced by the real world; they can just model it.
  - autoexec 2y ago
    > Mathematics is a tool that can be used to describe the real world, And that's what the article says the Greeks believed they could do using whole numbers and fractions. In the case of a real world square with sides of length 1 it seems that you can get away with describing the length of its diagonal in those terms which made it seem odd that it was what caused them to abandon their belief/aspiration. Maybe the author just described their dilemma poorly/strangely. I'm not at all suggesting that math has to be limited to the real world or that irrational numbers don't have their place. We're certainly better off with them.

nullc 2y ago

So then you notice that if you measure more precisely you get a different fraction... and even more precisely another different fraction. It's natural to ask what ratio you'd get if you measured with endless precision. In some cases there is such an answer and you can reason it out without conducting a series of increasingly precise measurements (though you could also find it with a sufficiently precise measurement), but in other cases there is no such answer. Isn't that interesting?