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>It doesn't make much sense to claim that real numbers are abstractions from quantities that exist as finite, quantized integers in empirical actuality. But if

>It doesn't make much sense to claim that real numbers are abstractions from quantities that exist as finite, quantized integers in empirical actuality. But if it's not an abstraction from something that clearly can be physically realized, then it is meaningful to ask whether a real-number quantity can exist in the physical universe. Real numbers are absolutely "physically realizable" (in the sense that you are defining it) in the Physical World. If you have 3 litres of water and you give me half, you have just "realized" the Real Number 1.5 from a "quantized integer" 3. Incidentally even integers are just an abstraction of attributes of collections of things i.e. cardinality of a set of things. This is why i tell people to look at Mathematics as a Formal System+Domain of Discourse in the Real World. You do all your symbol manipulations in the former and at the end map it to the real world to see whether it is valid.

jonathanstrange 3y ago
You're right that some real numbers are physically realizable, but not all of them. The debate usually focuses on irrational numbers. Please bear in mind this is an existing debate in the philosophy of mathematics, not my personal invention, and I have to apologize for being somewhat vague about. I tried to find a paper I've stumbled across years ago but couldn't find it, so I'm writing from distant memory. Anyway, the argument goes roughly like this: Real numbers also include the irrational numbers, and if these were physically realized, then they would contain an infinite amount of information within a finite space. This violates various physical laws. Now don't get me wrong, this is all controversial. The idea is, for example, that π cannot be physically realized because it has an infinite decimal expansion. Some people would agree, other would disagree. I understand why you disagree, but bear in mind my original point was not to argue that real numbers aren't physically real, but rather that there is no general agreement about this issue among people who muse about these kinds of philosophical questions. The question is relevant for foundational views about mathematics. If certain real numbers like 1/3 and π cannot be physically realized, they cannot be abstractions from something encountered in nature (at least not in the sense of "abstraction" according to which some properties are ignored). The view remains compatible with regarding them as mental constructions and compatible with mathematical Platonism, though.
- rramadass 3y ago
  The difficulties with Irrational Numbers were obvious from the beginning (https://en.wikipedia.org/wiki/Irrational_number#History https://en.wikipedia.org/wiki/Irrational_number#History). Many of them are very "Real" and occur naturally in the Universe (eg. Pi = Circumference of a circle / Diameter of the circle) but our modeling of them doesn't feel "natural" and hence the confusion. You might find the following interesting; 1) God created the Irrational Numbers : https://www.welovephilosophy.com/2014/03/26/god-created-the-irrational-numbers/ https://www.welovephilosophy.com/2014/03/26/god-created-the-... 2) What is a real-world metaphor for irrational numbers? : https://math.stackexchange.com/questions/2065998/what-is-a-real-world-metaphor-for-irrational-numbers https://math.stackexchange.com/questions/2065998/what-is-a-r... > Anyway, the argument goes roughly like this: Real numbers also include the irrational numbers, and if these were physically realized, then they would contain an infinite amount of information within a finite space. This violates various physical laws. See Do irrational numbers contain infinite information? : https://www.quora.com/Do-irrational-numbers-contain-infinite-information https://www.quora.com/Do-irrational-numbers-contain-infinite...
  - jonathanstrange 3y ago
    Well, Yes, that's what the debate is about. Just to make this clear (in case someone else reads this thread), this is an ongoing debate in the philosophy of mathematics and not just something people muse about on Quora and Stackexchange. What's important for me is that the notion of physical realizability is understood correctly. It does not pertain to philosophical arguments or common sense, it really means that something can be present as a quantity in nature (actuality) without violating existing physical laws. When someone argues that a mathematical structure or entity cannot be physically realized, the argument must concretely show that the realization would violate some currently well-confirmed laws of nature or fundamental physical principles. There is no disagreement with you. I just wanted to clarify that (hope you don't mind). I didn't have just any arguments against irrational numbers in mind but a specific type of arguments.
    - rramadass 3y ago

jonathanstrange 3y ago

You're right that some real numbers are physically realizable, but not all of them. The debate usually focuses on irrational numbers. Please bear in mind this is an existing debate in the philosophy of mathematics, not my personal invention, and I have to apologize for being somewhat vague about. I tried to find a paper I've stumbled across years ago but couldn't find it, so I'm writing from distant memory. Anyway, the argument goes roughly like this: Real numbers also include the irrational numbers, and if these were physically realized, then they would contain an infinite amount of information within a finite space. This violates various physical laws. Now don't get me wrong, this is all controversial. The idea is, for example, that π cannot be physically realized because it has an infinite decimal expansion. Some people would agree, other would disagree. I understand why you disagree, but bear in mind my original point was not to argue that real numbers aren't physically real, but rather that there is no general agreement about this issue among people who muse about these kinds of philosophical questions. The question is relevant for foundational views about mathematics. If certain real numbers like 1/3 and π cannot be physically realized, they cannot be abstractions from something encountered in nature (at least not in the sense of "abstraction" according to which some properties are ignored). The view remains compatible with regarding them as mental constructions and compatible with mathematical Platonism, though.

rramadass 3y ago
The difficulties with Irrational Numbers were obvious from the beginning (https://en.wikipedia.org/wiki/Irrational_number#History https://en.wikipedia.org/wiki/Irrational_number#History). Many of them are very "Real" and occur naturally in the Universe (eg. Pi = Circumference of a circle / Diameter of the circle) but our modeling of them doesn't feel "natural" and hence the confusion. You might find the following interesting; 1) God created the Irrational Numbers : https://www.welovephilosophy.com/2014/03/26/god-created-the-irrational-numbers/ https://www.welovephilosophy.com/2014/03/26/god-created-the-... 2) What is a real-world metaphor for irrational numbers? : https://math.stackexchange.com/questions/2065998/what-is-a-real-world-metaphor-for-irrational-numbers https://math.stackexchange.com/questions/2065998/what-is-a-r... > Anyway, the argument goes roughly like this: Real numbers also include the irrational numbers, and if these were physically realized, then they would contain an infinite amount of information within a finite space. This violates various physical laws. See Do irrational numbers contain infinite information? : https://www.quora.com/Do-irrational-numbers-contain-infinite-information https://www.quora.com/Do-irrational-numbers-contain-infinite...
- jonathanstrange 3y ago
  Well, Yes, that's what the debate is about. Just to make this clear (in case someone else reads this thread), this is an ongoing debate in the philosophy of mathematics and not just something people muse about on Quora and Stackexchange. What's important for me is that the notion of physical realizability is understood correctly. It does not pertain to philosophical arguments or common sense, it really means that something can be present as a quantity in nature (actuality) without violating existing physical laws. When someone argues that a mathematical structure or entity cannot be physically realized, the argument must concretely show that the realization would violate some currently well-confirmed laws of nature or fundamental physical principles. There is no disagreement with you. I just wanted to clarify that (hope you don't mind). I didn't have just any arguments against irrational numbers in mind but a specific type of arguments.
  - rramadass 3y ago