3 ms·

Sometimes math can be pretty surprising, such as when complex numbers might be shown to be necessary to explain quantum mechanics: https://www.nature.com/artic

by EGreg 3y ago

Sometimes math can be pretty surprising, such as when complex numbers might be shown to be necessary to explain quantum mechanics: https://www.nature.com/articles/s41586-021-04160-4 https://www.nature.com/articles/s41586-021-04160-4 Or when group theory predicts subatomic particles through symmetries: https://www.britannica.com/science/subatomic-particle/Hidden-symmetry https://www.britannica.com/science/subatomic-particle/Hidden... But other things like continuity and limits, as well as various topological spaces, seem to be purely mathematical constructs.

eigenket 3y ago
Continuity and limits, and some relevant topological spaces are absolutely required to describe very basic fundamental physics. You can't write down even stuff like Newton's laws without calculus, and you can't do calculus without something equivalent to limits.
rramadass 3y ago
See my other comment here: https://news.ycombinator.com/item?id=37656080 https://news.ycombinator.com/item?id=37656080 As the other commentator points out, "Continuity and Limits" are fundamental to explaining physical phenomena (via differential equations) and are not "purely mathematical constructs". PS: You might find this interesting; Imaginary Numbers are Real - https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703...
- jonathanstrange 3y ago
  You point to the argument that "we have to calculate with them, therefore they are real", but another aspect of the debate is about which mathematical entities are real in the sense of being physically realizable. From this perspective, it's not even clear that real numbers are real, e.g. whether and how it would physically possible for a real-number based quantity to be present in a finite space.
  - rramadass 3y ago
    If Mathematics helps in explaining Nature/Universe then it is "Real" no matter if the explanation goes through a whole stack of abstractions or not. There are two main parts; a) A Formal System consisting of Set of Objects, Operations, Mappings, Axioms and Logic Rules. We invent the symbols and notations to express these. b) A Domain of Discourse/Interpretation in which the above is applied to map to "Reality". We Humans have an innate sense of Quantities, Proportion, Objects and Relationships which is what can be called the "Mathematical sense". Even the most uneducated goatherd can count his goats (eg. using pattern-matching with one stone per goat) without knowing anything about the number system. He can also compare his bunch with his neighbour's and tell you which is larger. If you throw a ball at him he can estimate its trajectory and move accordingly to catch it. We have merely abstracted out the essentials from the above and modeled them as Set Theory, Integer/Real Number lines, Rate of change of one quantity w.r.t. another etc. and labeled these as "Mathematics". The models are by design "abstract" but once applied to a "domain" become concrete.
    - jonathanstrange 3y ago
      I agree with everything you say but that's not really what I was talking about. Physical realizability concerns the question whether some type of entity can in principle exist in the known physical universe, whether it's physical existence would violate existing laws of nature. The question is independent of the question whether there is (also) a Platonic realm of mathematical objects (although there is a connection if you are neither a constructivist nor a Platonist). As far as I know, nobody doubts that integer quantities can be physically realized without violating existing laws. Likewise, you can say that a square is an abstraction from a square macroscopic object, even though no side of that object can be perfectly square in nature. However, the case with real numbers is a bit different from the square. 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. From what I remember, some philosophers and physicists think the answer is No.

by EGreg 3y ago

eigenket 3y ago

Continuity and limits, and some relevant topological spaces are absolutely required to describe very basic fundamental physics. You can't write down even stuff like Newton's laws without calculus, and you can't do calculus without something equivalent to limits.

rramadass 3y ago

See my other comment here: https://news.ycombinator.com/item?id=37656080 https://news.ycombinator.com/item?id=37656080 As the other commentator points out, "Continuity and Limits" are fundamental to explaining physical phenomena (via differential equations) and are not "purely mathematical constructs". PS: You might find this interesting; Imaginary Numbers are Real - https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703...

jonathanstrange 3y ago
You point to the argument that "we have to calculate with them, therefore they are real", but another aspect of the debate is about which mathematical entities are real in the sense of being physically realizable. From this perspective, it's not even clear that real numbers are real, e.g. whether and how it would physically possible for a real-number based quantity to be present in a finite space.
- rramadass 3y ago
  If Mathematics helps in explaining Nature/Universe then it is "Real" no matter if the explanation goes through a whole stack of abstractions or not. There are two main parts; a) A Formal System consisting of Set of Objects, Operations, Mappings, Axioms and Logic Rules. We invent the symbols and notations to express these. b) A Domain of Discourse/Interpretation in which the above is applied to map to "Reality". We Humans have an innate sense of Quantities, Proportion, Objects and Relationships which is what can be called the "Mathematical sense". Even the most uneducated goatherd can count his goats (eg. using pattern-matching with one stone per goat) without knowing anything about the number system. He can also compare his bunch with his neighbour's and tell you which is larger. If you throw a ball at him he can estimate its trajectory and move accordingly to catch it. We have merely abstracted out the essentials from the above and modeled them as Set Theory, Integer/Real Number lines, Rate of change of one quantity w.r.t. another etc. and labeled these as "Mathematics". The models are by design "abstract" but once applied to a "domain" become concrete.
  - jonathanstrange 3y ago
    I agree with everything you say but that's not really what I was talking about. Physical realizability concerns the question whether some type of entity can in principle exist in the known physical universe, whether it's physical existence would violate existing laws of nature. The question is independent of the question whether there is (also) a Platonic realm of mathematical objects (although there is a connection if you are neither a constructivist nor a Platonist). As far as I know, nobody doubts that integer quantities can be physically realized without violating existing laws. Likewise, you can say that a square is an abstraction from a square macroscopic object, even though no side of that object can be perfectly square in nature. However, the case with real numbers is a bit different from the square. 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. From what I remember, some philosophers and physicists think the answer is No.