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> Isn't this article conflating our formalism of a given abstract entity (like real numbers or integers) with the abstract entity itself? It is arguing that th

> Isn't this article conflating our formalism of a given abstract entity (like real numbers or integers) with the abstract entity itself? It is arguing that the integers separate from the reals is the formalism and that the abstract entity is the reals. We also have a formalism of the reals, but it is closer to the abstract entity. > And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers! As we create better more useful formalisms, we interact more with the formalism than the thing itself. It is like putting on oven mitts to pick up a hot tray from the oven. This has already happened with the reals! Consider the question of if the reals are well-ordered or not. The question applies to the actual entity of the reals itself, but the absoluteness theorem shows that the question has no arithmetic consequences. You can simple ignore the question. Thus, those seeking the mental comfort and utility of the formalism do not have to concern themselves with the true nature of the real number line.

griffzhowl 1y ago
> It is arguing that the integers separate from the reals is the formalism and that the abstract entity is the reals. Why believe in abstract entities at all, as something distinct from the formalism? We have various formal or abstract concepts, that are useful in science and its applications, but they make contact with reality only through these uses: the natural numbers are used in counting, which is useful because there are many physical objects or events that are similar enough to each other that their differences can be neglected, then it's meaningful to count them and come up with a number for the collection, e.g. apples in a basket, planets in the solar system, or whatever. Real numbers developed from our perception of continuous physical magnitudes, and from the usefulness of applying the concept of number to these magnitudes. Then the formalism of the real numbers was developed based on mathematical constructions from the natural numbers/integers. We don't have to posit some abstract entity that the concept of real numbers refers to: it's a symbolic or mathematical construction that helps us in reasoning about (what we perceive as) continuous magnitudes, which aren't abstract, but concrete aspects of our experience of the world
- EthanHeilman 1y ago
  > Why believe in abstract entities at all, as something distinct from the formalism? Because they seem to have some actual reality independent of human ideas. Not that everyone agrees with this, but there are good arguments for it (see the last few thousand years of debates on the subject of platonic idealism).
  - griffzhowl 1y ago
    I'm saying that abstract entities only seem to have an independent existence because of a misconception that words function by referring to independent entities. Some of them do, but some also get their meaning from the part they play in a procedure, like the number words in the procedure of counting, and by extension the mathematical concepts built on these. There are reasons these particualr concepts are useful, but these reasons have to do with the structure of the concrete physical world and human activities in it, not an independent platonic reality. Wittgenstein writes on this theme quite a bit. The fact that something has been discussed for thousands of years also has nothing to do with whether there are good reasons for believing it, e.g. the Earth being flat or stationary at the center of the universe. People can be wrong for thousands of years
    - EthanHeilman 1y ago
      > The fact that something has been discussed for thousands of years also has nothing to do with whether there are good reasons for believing it, e.g. the Earth being flat or stationary at the center of the universe. People can be wrong for thousands of years I agree people can be wrong for thousands of years, but a flat Earth was conclusively disproved early on. I'm not talking about random people debating this issue, I'm talking a continuous intellectual process over thousands of years to understand the nature of mathematics. It may turn out that various theories put forward will be shown to be wrong. This has already happened, Godel's incompleteness was a major blow to people that argued that mathematics did not have an independent reality because it was just a logical game defined by axiomatic rules. Godel showed that axiomatic systems beyond a certain level of complexity have a reality beyond that logic can investigate.

griffzhowl 1y ago

> It is arguing that the integers separate from the reals is the formalism and that the abstract entity is the reals. Why believe in abstract entities at all, as something distinct from the formalism? We have various formal or abstract concepts, that are useful in science and its applications, but they make contact with reality only through these uses: the natural numbers are used in counting, which is useful because there are many physical objects or events that are similar enough to each other that their differences can be neglected, then it's meaningful to count them and come up with a number for the collection, e.g. apples in a basket, planets in the solar system, or whatever. Real numbers developed from our perception of continuous physical magnitudes, and from the usefulness of applying the concept of number to these magnitudes. Then the formalism of the real numbers was developed based on mathematical constructions from the natural numbers/integers. We don't have to posit some abstract entity that the concept of real numbers refers to: it's a symbolic or mathematical construction that helps us in reasoning about (what we perceive as) continuous magnitudes, which aren't abstract, but concrete aspects of our experience of the world

EthanHeilman 1y ago
> Why believe in abstract entities at all, as something distinct from the formalism? Because they seem to have some actual reality independent of human ideas. Not that everyone agrees with this, but there are good arguments for it (see the last few thousand years of debates on the subject of platonic idealism).
- griffzhowl 1y ago
  I'm saying that abstract entities only seem to have an independent existence because of a misconception that words function by referring to independent entities. Some of them do, but some also get their meaning from the part they play in a procedure, like the number words in the procedure of counting, and by extension the mathematical concepts built on these. There are reasons these particualr concepts are useful, but these reasons have to do with the structure of the concrete physical world and human activities in it, not an independent platonic reality. Wittgenstein writes on this theme quite a bit. The fact that something has been discussed for thousands of years also has nothing to do with whether there are good reasons for believing it, e.g. the Earth being flat or stationary at the center of the universe. People can be wrong for thousands of years
  - EthanHeilman 1y ago
    > The fact that something has been discussed for thousands of years also has nothing to do with whether there are good reasons for believing it, e.g. the Earth being flat or stationary at the center of the universe. People can be wrong for thousands of years I agree people can be wrong for thousands of years, but a flat Earth was conclusively disproved early on. I'm not talking about random people debating this issue, I'm talking a continuous intellectual process over thousands of years to understand the nature of mathematics. It may turn out that various theories put forward will be shown to be wrong. This has already happened, Godel's incompleteness was a major blow to people that argued that mathematics did not have an independent reality because it was just a logical game defined by axiomatic rules. Godel showed that axiomatic systems beyond a certain level of complexity have a reality beyond that logic can investigate.