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Does this represent the state of philosophy of mathematics, because I find this view quite naive. Mathematics is in the business of making a model of the real w

by jules 11y ago

Does this represent the state of philosophy of mathematics, because I find this view quite naive. Mathematics is in the business of making a model of the real world and then making falsifiable predictions with it, just like science. Take Fermat's last theorem: for all positive a,b,c and n>2, the value a^n + b^n - c^n never comes out 0. This is an empirically falsifiable statement. You can even make it a statement about real world objects if you wish: represent a number by a jar of coins, and do addition C=A+B by filling jar C with the coins in jar A and jar B together, multiplication AB by taking a whole jar B for each coin in A, etc. Rules of mathematical deduction are just devices for making predictions. So mathematical objects are neither "real" as in platonism, nor meaningless as in fictionalism.

roywiggins 11y ago
Embodied mathematics is what you are describing I think. The idea is that mathematics arises out of ordinary experience extended by metaphor. It charts a pretty satisfying middle ground between Platonism and Fictionalism. Mathematics is a story, but it is "at the bottom" tied into our experience of everyday reality. Do real numbers exist? Probably not, but the the continuum is like a long stick you can make marks on. Can you have a trillion apples, so you know that a trillion + a trillion is two trillion? No, but you can reason about "trillion" as existing metaphorically.
- hyperpape 11y ago
  That actually just sounds like good old fashioned nominalism or fictionalism--you'd have to sketch out a more detailed story, but I don't think it's actually distinct.
  - roywiggins 11y ago
    I'm (probably badly) summarizing Lakoff & Nunez's book: https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From Neither are capital-P Philosophers, and I don't remember if they actually make a claim that their ideas are philosophically distinct from fictionalism- they're more worried about platonists! The other approach I'm fond of is saying mathematical objects exist but only inside the heads of mathematicians. As long as mathematicians share a particular idea and can talk about it, it's math. So when I point to the real numbers, I'm pointing at a real thing: the shared idea of them. There's no Platonic realm, but they exist in the mind. It parallels how mathematics is really done (a proof is only a proof if it convinces mathematicians) so I find it compelling on that front.
danrisso3 11y ago
I pretty much agree with this position, although I find the "fill ... together" and "bring ... together" metaphors silly. For example, if you have two jars containing 50 plutionium coins each, putting all of them in the same jar will have some unforeseen consequences. However, this does not change the fact that there are 100 coins altogether.
- jules 11y ago
  It fails because you apply the model to a situation where it isn't applicable. This isn't very different than applying Newtonian physics with objects moving near the speed of light. By the way, numbers aren't defined as jars of coins. Numbers are a device to make predictions, and you can apply that device to make predictions about jars of coins. I think you find it silly because you know too much about numbers and have forgotten the early years when you learned about numbers. When you teach a little kid numbers you don't start with Peano's axioms, you start with jars of coins and then you build an abstract model and you convince the kid that the abstract model makes accurate predictions with a series of (thought) experiments.
GFK_of_xmaspast 11y ago
Be careful, that way lies ultra-finitism.
- jules 11y ago
  An interesting point but I'm not sure that it does. The ultra-finitist says that numbers above a certain limit don't exist. In my view numbers don't "exist" in the first place. The theory of numbers is a tool for making predictions. The statement that numbers above a certain limit don't exist turns into the statement that examples to test the theory don't exist above a certain limit, because you won't be able to find that big of an example in our universe. IMO this is a much more sensible statement. An ultra-finitist may worry about an intermediate number in a calculation not existing, even though the start and end number may be small (for instance consider log_2(2^n) = n). This is silly in my view, since the theory of numbers is a prediction tool as a whole, and there is no need to physically realize intermediate numbers. As long as you're using the rules of the theory you're good (for the same reason you don't need to be afraid of using infinite sets in the middle of a deduction). One might also worry that the theory of numbers may fail to make correct predictions for very large numbers, but this is really a different question about generalising from examples. It could already fail for very large examples that we can test in our universe. The possibility of it failing for numbers larger than we can realize in our universe is precisely what you don't need to worry about.
abrezas 11y ago
If a mathematical statement did not have a corresponding experimental test would you dismiss it? What about those?
- jules 11y ago
  All mathematical statements in constructive logic have an experimental test (actually since nonconstructive logic can be embedded in constructive logic via a double negation translation, they do too). Of course this isn't always the most interesting test, since usually you are modeling some higher level concept inside the logic, and you would have a more high level test. For example for the calculation of pi you would measure the ratio between the diameter and perimeter of a circle.
  - hyperpape 11y ago
    It sounds like we just stopped knowing a lot of the digits of pi if what you're saying is accurate: http://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/ http://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals...
    - jules 11y ago
      Pi isn't defined by how we measure it. It's defined mathematically. Predicting how a measurement of the ratio between the perimeter and the diameter of a circle comes out is just one of the predictions we can make having calculated pi inside mathematics.
      - abrezas 11y ago
        But if at some point further decimals cannot be experimentally proven to be right, at best you can say "pi is such and such for such precision" and any definition that gives the same number up to that precision must be accordingly accepted. Otherwise you need to demonstrate an experiment that uses further precision. But we don't do that, and that's why mathematics don't have to do with experience, they are an entirely different tool that also happens to be useful in experimental science.

by jules 11y ago

roywiggins 11y ago

Embodied mathematics is what you are describing I think. The idea is that mathematics arises out of ordinary experience extended by metaphor. It charts a pretty satisfying middle ground between Platonism and Fictionalism. Mathematics is a story, but it is "at the bottom" tied into our experience of everyday reality. Do real numbers exist? Probably not, but the the continuum is like a long stick you can make marks on. Can you have a trillion apples, so you know that a trillion + a trillion is two trillion? No, but you can reason about "trillion" as existing metaphorically.

hyperpape 11y ago
That actually just sounds like good old fashioned nominalism or fictionalism--you'd have to sketch out a more detailed story, but I don't think it's actually distinct.
- roywiggins 11y ago
  I'm (probably badly) summarizing Lakoff & Nunez's book: https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From Neither are capital-P Philosophers, and I don't remember if they actually make a claim that their ideas are philosophically distinct from fictionalism- they're more worried about platonists! The other approach I'm fond of is saying mathematical objects exist but only inside the heads of mathematicians. As long as mathematicians share a particular idea and can talk about it, it's math. So when I point to the real numbers, I'm pointing at a real thing: the shared idea of them. There's no Platonic realm, but they exist in the mind. It parallels how mathematics is really done (a proof is only a proof if it convinces mathematicians) so I find it compelling on that front.

danrisso3 11y ago

I pretty much agree with this position, although I find the "fill ... together" and "bring ... together" metaphors silly. For example, if you have two jars containing 50 plutionium coins each, putting all of them in the same jar will have some unforeseen consequences. However, this does not change the fact that there are 100 coins altogether.

jules 11y ago
It fails because you apply the model to a situation where it isn't applicable. This isn't very different than applying Newtonian physics with objects moving near the speed of light. By the way, numbers aren't defined as jars of coins. Numbers are a device to make predictions, and you can apply that device to make predictions about jars of coins. I think you find it silly because you know too much about numbers and have forgotten the early years when you learned about numbers. When you teach a little kid numbers you don't start with Peano's axioms, you start with jars of coins and then you build an abstract model and you convince the kid that the abstract model makes accurate predictions with a series of (thought) experiments.

GFK_of_xmaspast 11y ago

Be careful, that way lies ultra-finitism.

jules 11y ago
An interesting point but I'm not sure that it does. The ultra-finitist says that numbers above a certain limit don't exist. In my view numbers don't "exist" in the first place. The theory of numbers is a tool for making predictions. The statement that numbers above a certain limit don't exist turns into the statement that examples to test the theory don't exist above a certain limit, because you won't be able to find that big of an example in our universe. IMO this is a much more sensible statement. An ultra-finitist may worry about an intermediate number in a calculation not existing, even though the start and end number may be small (for instance consider log_2(2^n) = n). This is silly in my view, since the theory of numbers is a prediction tool as a whole, and there is no need to physically realize intermediate numbers. As long as you're using the rules of the theory you're good (for the same reason you don't need to be afraid of using infinite sets in the middle of a deduction). One might also worry that the theory of numbers may fail to make correct predictions for very large numbers, but this is really a different question about generalising from examples. It could already fail for very large examples that we can test in our universe. The possibility of it failing for numbers larger than we can realize in our universe is precisely what you don't need to worry about.

abrezas 11y ago

If a mathematical statement did not have a corresponding experimental test would you dismiss it? What about those?

jules 11y ago
All mathematical statements in constructive logic have an experimental test (actually since nonconstructive logic can be embedded in constructive logic via a double negation translation, they do too). Of course this isn't always the most interesting test, since usually you are modeling some higher level concept inside the logic, and you would have a more high level test. For example for the calculation of pi you would measure the ratio between the diameter and perimeter of a circle.
- hyperpape 11y ago
  It sounds like we just stopped knowing a lot of the digits of pi if what you're saying is accurate: http://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/ http://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals...
  - jules 11y ago
    Pi isn't defined by how we measure it. It's defined mathematically. Predicting how a measurement of the ratio between the perimeter and the diameter of a circle comes out is just one of the predictions we can make having calculated pi inside mathematics.
    - abrezas 11y ago
      But if at some point further decimals cannot be experimentally proven to be right, at best you can say "pi is such and such for such precision" and any definition that gives the same number up to that precision must be accordingly accepted. Otherwise you need to demonstrate an experiment that uses further precision. But we don't do that, and that's why mathematics don't have to do with experience, they are an entirely different tool that also happens to be useful in experimental science.