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I'm having trouble understanding the significance of Hartry Field's work. It sounds like he replaced the axiomatic system of what we call mathematics with an ax

I'm having trouble understanding the significance of Hartry Field's work. It sounds like he replaced the axiomatic system of what we call mathematics with an axiomatic system that he developed specifically for Newtonian gravity, and based on this he was able to claim that mathematics is dispensable. As an undergrad in logic class, I got the impression that mathematics is the discipline that studies axiomatic systems, so if you build an axiomatic system, you're doing mathematics. If that's true, then isn't all he did is just redefining the word mathematics in a very narrow sense and then dispensing with that narrowly defined notion?

hyperpape 11y ago
The question concerns the quantifiers that you see in math ("there exists an even prime number", "all numbers have a unique prime factorization"), etc. Field shows that for the theories necessary for Newtonian mathematics, we can treat those as quantifying over points in space-time, as opposed to anything distinctively mathematical. So we avoid Platonism. It sounds like you, like most people who haven't studied philosophy of math, assume some sort of pseudo-formalist account, and therefore don't find the question very compelling. Please don't take that disparagingly (most mathematicians are in the same boat, and I'm not sure I have much of an opinion at this point, save to note that people who haven't studied philosophy tend to think the issue is obvious in a way that a lot of philosophers don't).
- samth 11y ago
  I think most mathematicians are more likely to be Platonists than formalists. Formalism is also a real position in philosophy of math, but a minority one for I think two reasons. One is that the math we've developed seems significant enough and effective enough to deserve more explanation than just "it's one of many possible formal systems". Second, if you take a straightforward formalist approach, then there aren't really any interesting questions in philosophy of math, so people who think that are less likely to become philosophers of math.
  - cokernel 11y ago
    It's possible that a mathematician with philosophical commitments is more likely to be a Platonist than a formalist, but I suspect that most mathematicians are neither ("Philosophy is for the philosophers"). The so-called "naive" view that most mathematicians may be described as having ("I'm discovering something real when I prove a theorem", "Mathematics is objective", and so on) certainly sounds similar to the language used by Platonists, but I have yet to run into a mathematician who is comfortable with the logical conclusions of a strict Platonism. In particular, there are epistemological problems with Platonism (how can we know anything about non-physical entities?) that cause problems in the justification of mathematics and would be nice to be able to ignore.
    - samth 11y ago
      I think we mostly agree on the empirical issues -- mathematicians usually don't care about the philosophical issues, and espouse naive Platonism which they haven't thought through. However, I don't think it's right to say that they're formalist just because they wouldn't agree, upon reflection, with Platonism's conclusions. I (like a lot of computer scientists) am a formalist, but it's not fair to say that other people secretly agree with me because I think Platonism has major flaws.
- KKKKkkkk1 11y ago
  I'm just having trouble understanding what is the difference between the points of a suitable geometrical space and the points of SPACETIME (other than the suggestive name). I guess that this difference is key to philosophers while most mathematicians and physicists would fail to grasp it.
  - hyperpape 11y ago
    I can kick (some) things in spacetime.

hyperpape 11y ago

The question concerns the quantifiers that you see in math ("there exists an even prime number", "all numbers have a unique prime factorization"), etc. Field shows that for the theories necessary for Newtonian mathematics, we can treat those as quantifying over points in space-time, as opposed to anything distinctively mathematical. So we avoid Platonism. It sounds like you, like most people who haven't studied philosophy of math, assume some sort of pseudo-formalist account, and therefore don't find the question very compelling. Please don't take that disparagingly (most mathematicians are in the same boat, and I'm not sure I have much of an opinion at this point, save to note that people who haven't studied philosophy tend to think the issue is obvious in a way that a lot of philosophers don't).

samth 11y ago
I think most mathematicians are more likely to be Platonists than formalists. Formalism is also a real position in philosophy of math, but a minority one for I think two reasons. One is that the math we've developed seems significant enough and effective enough to deserve more explanation than just "it's one of many possible formal systems". Second, if you take a straightforward formalist approach, then there aren't really any interesting questions in philosophy of math, so people who think that are less likely to become philosophers of math.
- cokernel 11y ago
  It's possible that a mathematician with philosophical commitments is more likely to be a Platonist than a formalist, but I suspect that most mathematicians are neither ("Philosophy is for the philosophers"). The so-called "naive" view that most mathematicians may be described as having ("I'm discovering something real when I prove a theorem", "Mathematics is objective", and so on) certainly sounds similar to the language used by Platonists, but I have yet to run into a mathematician who is comfortable with the logical conclusions of a strict Platonism. In particular, there are epistemological problems with Platonism (how can we know anything about non-physical entities?) that cause problems in the justification of mathematics and would be nice to be able to ignore.
  - samth 11y ago
    I think we mostly agree on the empirical issues -- mathematicians usually don't care about the philosophical issues, and espouse naive Platonism which they haven't thought through. However, I don't think it's right to say that they're formalist just because they wouldn't agree, upon reflection, with Platonism's conclusions. I (like a lot of computer scientists) am a formalist, but it's not fair to say that other people secretly agree with me because I think Platonism has major flaws.
KKKKkkkk1 11y ago
I'm just having trouble understanding what is the difference between the points of a suitable geometrical space and the points of SPACETIME (other than the suggestive name). I guess that this difference is key to philosophers while most mathematicians and physicists would fail to grasp it.
- hyperpape 11y ago
  I can kick (some) things in spacetime.