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I'm going to ramble a bit. I haven't thoroughly philosophically investigated any of this, this is just a condensed version of my train of thought: There is a v

I'm going to ramble a bit. I haven't thoroughly philosophically investigated any of this, this is just a condensed version of my train of thought: There is a very obvious way in which mathematics is related to nature: the concept of an 'integer' originates directly from nature. The integers initially allowed our ancestors to count predators, prey, enemies. They did not originate from some brilliant flash of abstract light. The abstract definition of the integers is merely an ad hoc construction to make some things easier. They were already known and used, very effectively, in ancient times, without that abstraction. They are only known to us, only came into existence as memes in our memepool, because they are useful for this purpose. Integers count multiplicity: how many distinct objects does a group of objects, considered 'equal' for the purposes for which you are counting them, contain? Sometimes just knowing they are unequal is enough, making integers themselves signifiers of a difference. Sometimes you want to know how unequal they are. Do I need to kill one other bison to have the same amount as we had yesterday? Twice as many (meaning: the same count once over)? Everything else is just derivative from those principles: counting things considered equal and the equality of the countings themselves. Everything else in mathematics is just convenient representations of Gödel numberings to say what is equal and how much things aren't. So of course mathematics applies to nature. Mathematics was born as a way to abstract over nature, to quantify it.

yequalsx 12y ago
I think it is definitely not true that "<i>Everything else in mathematics is just convenient representations of Gödel numberings to say what is equal and how much things aren't</i>" It is true that it appears that non-negative integers are inherent from nature. Things come in discretized quantities. From non-negative integers one gets in an isomorphically unique way all integers. From these we get in an isomorphically unique way the rationals. The completion of the rationals are the reals. The algebraic closure of the reals are the complex numbers. At each stage the extension is unique. So in some sense complex numbers are baked into the universe. What is surprising, at least to some, is that these concepts are useful in making predictions about the universe and in describing how the universe works.
- Confusion 12y ago
  Well, suppose the following: Given a universe finite in extent, with a finite lifetime and a finite mass-energy content, then for all possible practical purposes only reals up to a finite precision are needed to calculate and describe observable properties of the universe. With that constraint, there is e.g. an isomorphism between integers and all needed reals and as such every observable property in the universe can be quantified using the integers. It may seem that that would 'invalidate' a lot of mathematical machinery, which uses concepts that can not be enumerated by the integers (i.e. the set of reals), but that need not be the case. The mathematical machinery we have developed may partly be a very effective 'thought experiment': what if the reals were uncountable? It turns out that makes things easier to calculate than to take their countability into account, while giving results indistinguishable from taking their countability into account. I lack the mathematical sophistication to prove this is possible. I'm just proposing a view of mathematics that would make it obvious why it is effective. I don't believe in miracles or coincidence and I'm not a Platonist or other sort of idealist, so that doesn't leave me much choice, does it? :)
  - yequalsx 12y ago
    Suppose you get into your car and start driving. You drive for 1 hour and travel 70 miles. At the end of 1 hour your car is at rest since you've reached your destination. Now let x be a number less than or equal to 70. Was there a point in time you were traveling x miles per hour? Is this true for all x in [0, 70]? If so then there is a need for an uncountable number of reals. If not then which values of x did you skip? Do these values of x become part of the needed numbers since they are an excluded set of a numbers of an experiment?
    - judk 12y ago
      In the first branch, you are assuming the existence of uncountably continuous time and space with perfect precision. Scientific observation is not on your side. (Plancks constant, QM, for example) For the second, if the universe were somehow rational, there is no need to drag in irrational numbers to describe it.

yequalsx 12y ago

I think it is definitely not true that "<i>Everything else in mathematics is just convenient representations of Gödel numberings to say what is equal and how much things aren't</i>" It is true that it appears that non-negative integers are inherent from nature. Things come in discretized quantities. From non-negative integers one gets in an isomorphically unique way all integers. From these we get in an isomorphically unique way the rationals. The completion of the rationals are the reals. The algebraic closure of the reals are the complex numbers. At each stage the extension is unique. So in some sense complex numbers are baked into the universe. What is surprising, at least to some, is that these concepts are useful in making predictions about the universe and in describing how the universe works.

Confusion 12y ago
Well, suppose the following: Given a universe finite in extent, with a finite lifetime and a finite mass-energy content, then for all possible practical purposes only reals up to a finite precision are needed to calculate and describe observable properties of the universe. With that constraint, there is e.g. an isomorphism between integers and all needed reals and as such every observable property in the universe can be quantified using the integers. It may seem that that would 'invalidate' a lot of mathematical machinery, which uses concepts that can not be enumerated by the integers (i.e. the set of reals), but that need not be the case. The mathematical machinery we have developed may partly be a very effective 'thought experiment': what if the reals were uncountable? It turns out that makes things easier to calculate than to take their countability into account, while giving results indistinguishable from taking their countability into account. I lack the mathematical sophistication to prove this is possible. I'm just proposing a view of mathematics that would make it obvious why it is effective. I don't believe in miracles or coincidence and I'm not a Platonist or other sort of idealist, so that doesn't leave me much choice, does it? :)
- yequalsx 12y ago
  Suppose you get into your car and start driving. You drive for 1 hour and travel 70 miles. At the end of 1 hour your car is at rest since you've reached your destination. Now let x be a number less than or equal to 70. Was there a point in time you were traveling x miles per hour? Is this true for all x in [0, 70]? If so then there is a need for an uncountable number of reals. If not then which values of x did you skip? Do these values of x become part of the needed numbers since they are an excluded set of a numbers of an experiment?
  - judk 12y ago
    In the first branch, you are assuming the existence of uncountably continuous time and space with perfect precision. Scientific observation is not on your side. (Plancks constant, QM, for example) For the second, if the universe were somehow rational, there is no need to drag in irrational numbers to describe it.