It would help if you told us what you think the mathematical definition of zero is.
That’s a thread entirely about mathematics. I’d like to know what part of physics you find it difficult to deal with zero in that isn’t subtracting one apple from another, which isn’t a physics problem I’ve ever encountered before.
BTW, I’m asking other posters please not to supply a mathematical definition of zero. We can’t get to the source of the OP’s confusion unless they specify their own thinking.
I’ve also seen it questioned sometimes whether there is really “infinite” of anything in the natural universe, or whether the whole concept of infinity is purely an abstract concept and nothing more.
Is there a mathematical definition of zero (short of going into definitions of subatomic granularity)?
In geometry, a “point” is undefined. It is understood that you can only define things in terms of other things that are already defined, thus you have to start somewhere with a few undefined terms. Likewise, facts are proved via arguments that use prior known facts as premises, thus you have to start somewhere with a few unproven facts (axioms).
Isn’t there something similar in algebra? We have some axioms. Aren’t there also some undefined terms also?
If you develop your algebra in terms of set theory so that your algebraic structures are sets, then zero is an element of your set, satisfying certain properties. I assume algebra is still formally taught this way. At some point they typically mention a “Grothendieck universe”.
Sally has 7 apples. She gives Tom 5 of them, and she gives Helen 2 of them. How many apples does Sally have left?
Answer: 7 - (5 + 2) = 0
@Richard_Pearse I heard an interesting distinction once between zero and nothing as follows: If I have a bank account with a balance of $100, and withdraw $100, the balance in the account is $0. If I don’t have a bank account, the balance doesn’t exist, and so is nothing.
Maybe also getting a bit towards the OP, I would say that at its fundemental level math is really just set of rules for symbol manipulation. Using these rules you can write sequences of symbols that follow those rules and if by following those rules you are able to write a particular elegant set of symbols than other mathematicians become very happy and invite you to their ivory towers to talk about the special ways you manipulated your symbols and may even give you modest amounts of money. Why on earth do they do this?
Now what physics has discovered, is that there is a correspondence between the sets of symbols you can write down following the rules and observations you make in the real world. You can make an observation about the real world, translate it into symbols, manipulate those symbols according to the rules, and then when you look at what you end up with, by some miracle it matches what you later observe. This miracle always seems to hold and in cases where it doesn’t hold physicists get very excited and work out new different ways to translate the observations into symbols are invented so that it works again.
That this symbol manipulation and translation allows one to predict and manipulate reality. This is why people are willing to pay people some people money to manipulate symbols all day, and other people money to make observations and see how well they translate into symbols.
One of the symbols mathematicians manipulate is zero. Depending on the application it gets translated by physicists into a variety of different observations, and works really really well.
Probably also need them to offer their physics definition as well. If the issue is definitions.
My WAG though is that the issue is what they consider abstractions vs reality: like a perfect circle or an infinitely small point being possible and useful as mathematical abstractions but when applied to reality are useful metaphors without exact physical analogs. We treat real world items as if they are these abstractions but they are not quite.
Of course though in physics sometimes those abstractions have a basis in reality - as mentioned already one fundamental particle of a certain type is exactly the same as another of that type differing only by where/when it is and where it is going.
I don’t think that’s a problem with definitions per se. It is an issue with how we use abstract concepts as tools to describe reality that is messier than ideal forms.
Thank you for this. You’ve articulated my question much better than I did with apples.
This is an interesting wiki article on zero-point energy..
Zero isn’t unique in this aspect though, the same issue applies to other numbers.
You’re welcome but I still don’t get what your exact question is.
That can be a start of a discussion sure. But what is the question?
If you say that your space-time is a real manifold, say to set up a quantum field theory, it does not necessarily mean that you believe the mathematical real numbers physically exist in some philosophical sense. But when physicists talk about zero as a real number or as an integer, they do mean the same thing as mathematicians. [As far as I can tell!?] That is the nature of theoretical physics, and, while there is a lot to work on, the issues are not really with simple mathematics like if a particle has spin zero, is that really zero. If there is any kind of problem there, it is with your quantum theory, not with representation theory.
Except that if your balance falls to lower than $100, the bank charges a $3/month maintenance fee. So your balance is going to be -$3 soon.
Well, then you charge them the “you’re a shitty bank fee” and move your $0 somewhere else.
The intent of my OP was to learn about a question I’ve had since I learned algebra; how can the concepts of zero and infinity apply to natural sciences? I apologize if my question isn’t exact. But I’m learning.
Do you feel your question was answered?
It isn’t only that they can be applied to the natural sciences, developing models for the natural sciences with explanatory and predictive power depend on those concepts.The lack of zero hobbled the Western world for centuries and its invention and then importation to the West sparked the birth of maths that allowed maths to be the useful tools, to have the descriptive powers, that they now have.
You may find this article of interest:
I didn’t expect that my inexact question would have an exact answer. Thanks for the Vox link. My sister teaches 7th grade math, so she might like it.
This interests me: