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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? Surely quantities existe

Isn't this article conflating our formalism of a given abstract entity (like real numbers or integers) with the abstract entity itself? Surely quantities existed long before humans (e.g. there was a quantity of stars in the Milky Way 1 million years ago). And surely ordinals existed long before humans (e.g. there was a most massive star in the Milky Way 1 million years ago). The article's claim seems to be about the mathematical formalisms humans have invented for integers and real numbers. 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!

john-h-k 1y ago
> 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! Everything you can express in integers you can express in reals, but there are many things expressable in reals not possible in integers. It would be surprising if the formalism for a thing that completely supersets another thing had an equally simple formalism
- orlp 1y ago
  Actually in math it's very common for the more general system to be simpler. Compare for example the prime numbers with the integers, or general groups with finite simple groups and the monster group.
- tshaddox 1y ago
  I'm not suggesting the two formalisms should be equally simple. But surely it's not controversial to claim that formalizing the reals involves much more advanced mathematics (and runs into much deeper problems) than formalizing the integers. I'd argue that this disparity is slightly surprising, given that both the integers and the reals are ubiquitous in essentially all branches and levels of mathematics.
- xscott 1y ago
  > [...] but there are many things expressible in reals not possible in integers Are you sure there is anything we can express in the Reals that isn't an integer in disguise? The first answer might be the sqrt(2) or pi, but we can write a finite program to spit out digits of those forever (assuming a Turing machine with integer positions on a countably infinite length tape). The binary encoding of the program represents the number, and it only needs to be finite, not even an integer at infinity. Then you might say Chaitin's constant, but that's just a name for one value we don't know and can't figure out. You can approximate it to some number of digits, but that doesn't seem good enough to express it. You can prove a program can't emit all the digits indefinitely. And even if you could, is giving one Real number a name enough? Names are countable, and again arguably finite. It seems to me we can prove there are more Reals than there are Integer or Computable numbers, but we can't "express" more than a finite number of those which aren't computable. Integers in disguise.
  - tshaddox 1y ago
    It seems like you're suggesting that mathematicians replace the reals with the computables. This is a reasonable thing to try, and is likely of particular interest to constructivists. There's even this whole field: https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis
    - xscott 1y ago
      > It seems like you're suggesting that mathematicians replace the reals with the computables. No, not any more. :-) I liked the idea a year or two ago, but I've come to believe that even the Integers are too bizarre to worry about. For now, I'm content just playing with fixed and floating point, maybe with arbitrary precision. Stuff I can reason about. I just think people are a little too casual thinking they are really using the Reals. It might be like Feynman's quote about saying you understand QM.
crazygringo 1y ago
I mean, this is one of the deepest questions of philosophy. Do or can concepts and categories exist without the beings that create them? Does tuna casserole exist independently of humans? If not, how is the idea of the number 7 different from the idea of a tuna casserole? Or what about the concept of decision by majority, which isn't as basic as 7, but doesn't have the physicality of a casserole?
- 1718627440 1y ago
  > Do or can concepts and categories exist without the beings that create them No of course not, but that's not the question. The question is whether the concepts were created by the beings. In a system with 7 stars the number of stars doesn't change when all humans die or humans never even developed.
  - crazygringo 1y ago
    If humans never even developed, many philosophers would say the concepts of "number" and "7" and "star" would not exist. Others would say they all exist independently of humans. Some think abstract mathematical concepts are more privileged than physical categories like stars. That is the question. I can't even tell if you're arguing one side or the other because you say "of course not" followed by "doesn't change" when those appear to be contradictory positions. But of course there are a lot of subtleties here. There's no "of course" to any of it. These things take entire books to argue one side and try to refute the other.
    - 1718627440 1y ago
      > many philosophers If only this was the stuff humans wasted their energy on not shooting explosives to improve their ego. :-) > If humans never even developed, many philosophers would say the concepts of "number" and "7" and "star" would not exist. Yes, I'm in the camp that thinks this is just plain wrong. > abstract mathematical concepts are more privileged than physical categories like stars. I agree to that. > I can't even tell if you're arguing one side or the other Sorry, if I was unclear. You wrote: > Do or can concepts and categories exist without the beings that create them? This takes it a priori, that the categories are created by the beings. If this is true, then I think by definition this categories can't precede the beings. That's why I wrote "of course not". When things are created by beings, they don't precede them, when they are not, they do. What I already disagree with is what you already implied as a given. Honestly I don't even know how to argue for that, because this is what I think is part of the definition of the concepts and categories. When I coin some term it always happens in reaction to things I perceived with my senses. So given my senses don't lie to me, of course this is outside of me. There are also things that are in fact invented by humans, for example color names. This doesn't mean that colors don't exist independently, but the exact boundary is arbitrary. There are also concepts where I think they can be both. For example beauty. When applied to skin or shape, these are just invented, but the beauty of complexity or completeness exists outside of us. And beauty in general only exists as it points as to a thing that exists independently of us.
EthanHeilman 1y ago
> 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
  - 1y ago
lotharcable 1y ago
Modern science is derived from Christian Scholasticism from the middle ages so this way of talking and thinking about science as being divinely originated is only unusual in the past couple centuries or so. It is from that era that they developed systems of rigorous debate, formal logic, and things like peered reviewed papers that we call "the scientific method". As far as the history of these sorts of mathematical discussions the concept of negative numbers didn't exist until the 15 century. I am sure that each new concept was faced with some resistance and debate on its true nature before it became widely accepted. So I am sure that somebody looking through the historical record could find all sorts of wild quotes from different theologians trying to grasp new concepts and reconcile them with existing mathmatical standards.
- griffzhowl 1y ago
  > Modern science is derived from Christian Scholasticism from the middle ages No, I don't think so. It seems much more based on ancient Greek geometry and logic, the Indian numeral system and Arabic algebra. Modern science really took off after Galileo, at the time when the ancient Greek works were recovered in Europe and could be synthesized with the arithmetic and algebra of these other cultures. Galileo himself credits the "divus" Archimedes as his main inspiration. What aspects of Christian scholasticism do you think developed into modern science?
andrewla 1y ago
> a quantity of stars There is no fundamental unit of "star". Maybe we can talk about electrons or protons or something, but what is and is not a star is a model, not a reality. Concretely, a bundle of pre-stellar gasses at some point transitions to being a star, but when in that time spectrum does it make that transition? When in the process of stellar exhaustion does it stop being a star?

john-h-k 1y ago

> 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! Everything you can express in integers you can express in reals, but there are many things expressable in reals not possible in integers. It would be surprising if the formalism for a thing that completely supersets another thing had an equally simple formalism

orlp 1y ago
Actually in math it's very common for the more general system to be simpler. Compare for example the prime numbers with the integers, or general groups with finite simple groups and the monster group.
tshaddox 1y ago
I'm not suggesting the two formalisms should be equally simple. But surely it's not controversial to claim that formalizing the reals involves much more advanced mathematics (and runs into much deeper problems) than formalizing the integers. I'd argue that this disparity is slightly surprising, given that both the integers and the reals are ubiquitous in essentially all branches and levels of mathematics.
xscott 1y ago
> [...] but there are many things expressible in reals not possible in integers Are you sure there is anything we can express in the Reals that isn't an integer in disguise? The first answer might be the sqrt(2) or pi, but we can write a finite program to spit out digits of those forever (assuming a Turing machine with integer positions on a countably infinite length tape). The binary encoding of the program represents the number, and it only needs to be finite, not even an integer at infinity. Then you might say Chaitin's constant, but that's just a name for one value we don't know and can't figure out. You can approximate it to some number of digits, but that doesn't seem good enough to express it. You can prove a program can't emit all the digits indefinitely. And even if you could, is giving one Real number a name enough? Names are countable, and again arguably finite. It seems to me we can prove there are more Reals than there are Integer or Computable numbers, but we can't "express" more than a finite number of those which aren't computable. Integers in disguise.
- tshaddox 1y ago
  It seems like you're suggesting that mathematicians replace the reals with the computables. This is a reasonable thing to try, and is likely of particular interest to constructivists. There's even this whole field: https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis
  - xscott 1y ago
    > It seems like you're suggesting that mathematicians replace the reals with the computables. No, not any more. :-) I liked the idea a year or two ago, but I've come to believe that even the Integers are too bizarre to worry about. For now, I'm content just playing with fixed and floating point, maybe with arbitrary precision. Stuff I can reason about. I just think people are a little too casual thinking they are really using the Reals. It might be like Feynman's quote about saying you understand QM.

crazygringo 1y ago

I mean, this is one of the deepest questions of philosophy. Do or can concepts and categories exist without the beings that create them? Does tuna casserole exist independently of humans? If not, how is the idea of the number 7 different from the idea of a tuna casserole? Or what about the concept of decision by majority, which isn't as basic as 7, but doesn't have the physicality of a casserole?

1718627440 1y ago
> Do or can concepts and categories exist without the beings that create them No of course not, but that's not the question. The question is whether the concepts were created by the beings. In a system with 7 stars the number of stars doesn't change when all humans die or humans never even developed.
- crazygringo 1y ago
  If humans never even developed, many philosophers would say the concepts of "number" and "7" and "star" would not exist. Others would say they all exist independently of humans. Some think abstract mathematical concepts are more privileged than physical categories like stars. That is the question. I can't even tell if you're arguing one side or the other because you say "of course not" followed by "doesn't change" when those appear to be contradictory positions. But of course there are a lot of subtleties here. There's no "of course" to any of it. These things take entire books to argue one side and try to refute the other.
  - 1718627440 1y ago
    > many philosophers If only this was the stuff humans wasted their energy on not shooting explosives to improve their ego. :-) > If humans never even developed, many philosophers would say the concepts of "number" and "7" and "star" would not exist. Yes, I'm in the camp that thinks this is just plain wrong. > abstract mathematical concepts are more privileged than physical categories like stars. I agree to that. > I can't even tell if you're arguing one side or the other Sorry, if I was unclear. You wrote: > Do or can concepts and categories exist without the beings that create them? This takes it a priori, that the categories are created by the beings. If this is true, then I think by definition this categories can't precede the beings. That's why I wrote "of course not". When things are created by beings, they don't precede them, when they are not, they do. What I already disagree with is what you already implied as a given. Honestly I don't even know how to argue for that, because this is what I think is part of the definition of the concepts and categories. When I coin some term it always happens in reaction to things I perceived with my senses. So given my senses don't lie to me, of course this is outside of me. There are also things that are in fact invented by humans, for example color names. This doesn't mean that colors don't exist independently, but the exact boundary is arbitrary. There are also concepts where I think they can be both. For example beauty. When applied to skin or shape, these are just invented, but the beauty of complexity or completeness exists outside of us. And beauty in general only exists as it points as to a thing that exists independently of us.

EthanHeilman 1y ago

> 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
- 1y ago

lotharcable 1y ago

Modern science is derived from Christian Scholasticism from the middle ages so this way of talking and thinking about science as being divinely originated is only unusual in the past couple centuries or so. It is from that era that they developed systems of rigorous debate, formal logic, and things like peered reviewed papers that we call "the scientific method". As far as the history of these sorts of mathematical discussions the concept of negative numbers didn't exist until the 15 century. I am sure that each new concept was faced with some resistance and debate on its true nature before it became widely accepted. So I am sure that somebody looking through the historical record could find all sorts of wild quotes from different theologians trying to grasp new concepts and reconcile them with existing mathmatical standards.

griffzhowl 1y ago
> Modern science is derived from Christian Scholasticism from the middle ages No, I don't think so. It seems much more based on ancient Greek geometry and logic, the Indian numeral system and Arabic algebra. Modern science really took off after Galileo, at the time when the ancient Greek works were recovered in Europe and could be synthesized with the arithmetic and algebra of these other cultures. Galileo himself credits the "divus" Archimedes as his main inspiration. What aspects of Christian scholasticism do you think developed into modern science?

andrewla 1y ago

> a quantity of stars There is no fundamental unit of "star". Maybe we can talk about electrons or protons or something, but what is and is not a star is a model, not a reality. Concretely, a bundle of pre-stellar gasses at some point transitions to being a star, but when in that time spectrum does it make that transition? When in the process of stellar exhaustion does it stop being a star?