The discussion centers on the physical meanings of rational, irrational, and complex numbers, concluding that only whole numbers (0, 1, 2, 3, etc.) have direct physical interpretations when associated with tangible objects. Participants argue that while numbers can describe physical quantities, their mathematical nature is abstract and independent of physical reality. For instance, the number -5 lacks physical meaning unless contextualized, such as in temperature (-5°C) or direction (-5 m). The distinction between rational and irrational numbers is deemed irrelevant in many physical contexts, emphasizing that numbers serve as mathematical objects rather than models of physical phenomena.
PREREQUISITESMathematicians, physicists, educators, and students interested in the foundational concepts of numbers and their applications in both mathematics and physical sciences.
One can argue that even whole numbers are not necessarily "physical". Counting depends upon being able to distinguish things. Some things, like elephant, can be counted easily but, say, slime mold. Can you always say that there are a specific number of slime mold in a yard?Elucidus said:The only numbers that one could argue have "physical meaning" are the whole numbers 0, 1, 2, 3, etc. I have 1 tree in my backyard, I am instructing 4 courses this semester, I own 5 pets, and 0 rockets, etc.
All other numeric constructions, although useful, do not have a "phyiscal meaning". If you take a piece of chalk and break it in half, you have two pieces of chalk (not a fraction). What would be the "physical meaning" of the number -5?
The more outre a number gets, the more it is removed from a physical sense. Complex numbers, quarternions, octonions, etc all are very abstract and it would be extremely difficult to fabricate a reality for them.
--Elucidus
You gave these numbers a physics meaning when you associated them with objects, such as trees. The numbers on their own have no physical meanings. For example, the number -5 has no physical meaning, as you correctly assert. However, if I were to say -5oC, that certainly has a physical meaning. Equally, the natural numbers do not have a physically meaning until one them with associates a physical quantity.Elucidus said:The only numbers that one could argue have "physical meaning" are the whole numbers 0, 1, 2, 3, etc. I have 1 tree in my backyard, I am instructing 4 courses this semester, I own 5 pets, and 0 rockets, etc.
Hootenanny said:You gave these numbers a physics meaning when you associated them with objects, such as trees. The numbers on their own have no physical meanings. For example, the number -5 has no physical meaning, as you correctly assert. However, if I were to say -5oC, that certainly has a physical meaning. Equally, the natural numbers do not have a physically meaning until one them with associates a physical quantity.
Numbers, whether real, imaginary, complex, rational, irrational, natural or otherwise are not 'mathematical models', they are simply mathematical objects. Granted, it is likely that the concept of numerals and numbers originated with the need to quantify a group of physical objects. However, our current definition of numbers is completely independent of any physical interpretation. As I said earlier and dx also comments, one can associate a physical meaning with any set of numbers, but that doesn't mean that that, or any other physical interpretation, is an inherent property of that set of numbers.S_David said:But there is a strong relationship between physical and mathematical models, where the latter used to describe the former in a more convenient way. For example, one can say that -5 m is traveling of an object in an opposite (presumed) direction, the same is applied of -5 Ampere, where the minus sign here describes direction.