Is zero in mathematics the same as zero in physics?

Factual Questions
65

So in that example let’s consider a particle like an electron satisfying the Schrodinger equation. So already there is the question of how realistic that is. There should be some rotational symmetry, in this case SU(2). But if we have gotten this far it is a relatively simple exercise to work out the irreducible representations of SU(2) and see that spin must be quantized so that the spin angular momentum can only be \hbar\sqrt{s(s+1)} for s = 0,\,\frac{1}{2},1,\frac{3}{2},\ldots

Seems like bullshit? Ridiculed by Pauli as having “nothing to do with reality”? Nobel Prize for Otto Stern, though.

66

This is an interesting link about Otto Stern.

Also watching my march madness bracket.

67

“Apple” isn’t a unit in math nor in physics. When you say “one apple minus one apple”, you are expressing “quantity 1 minus quantity 1”. You’ve simply chosen to represent the quantity in apples for (I guess) some sort of concrete demonstration for teaching purposes, but the actual math doesn’t require it.

The reason it seems different in physics is that physics (at least classical physics) usually deals with magnitudes instead of quantities, but the math is the same. 1 gram minus 1 gram is zero grams.

68

There was a story somewhere, some years ago, about a guy who tried to do some banking business and discovered that his bank account had mysteriously ceased to exist, vanished without a trace. Even the bank employees took a long time to figure out what had happened.

It turned out, he had written a check for exactly the amount in his account which, when it cleared, reduced his balance to $0.00. It further turned out, that in this bank’s computer implementation, setting the balance to zero was the way an account was closed. If he had overdrawn and had a negative balance this would not have happened. But withdrawing exactly the total balance, as in this case, closed the account.

So, fortuitously, no real harm was done because he lost this account only when the balance was 0 so he didn’t lose any money. Still, as I recall the story, he went to a different bank after that and took his $0.00 there.

Here’s another true story that actually happened to me: Once upon a time, I closed an account and took my money elsewhere, for whatever reason. This was in the before-times when bank accounts could generate some non-trivial amount of interest. Later, I moved. So I called the bank of the now-closed account to give them my new mailing address.

The rep I spoke to said they couldn’t take an address change for an account that no longer existed, and wondered why I thought they would need my new address.

Well, duh. Account records continue to exist even after an account is closed, because reasons. In particular, I pointed out, they would need to send me a 1099 at the end of the year. Thus explained, the rep suddenly discovered that yes, they could take an address update for the non-existing account. (Because the account actually still exists, even in its closed status.)

69

Using an apple as a unit was not a good idea. I blame Issac Newton for my mistake. My intention in making the OP was to learn how mathematics can describe the natural world, and there are some very interesting posts and links in this thread.

70

Mathematics is a tool used by physicists to describe the natural world. The zero used in physics is the same zero used mathematics because it is being used within mathematics itself. That said, any answer provided by an equation in physics is only as accurate as the accuracy of the input and will therefore be an approximation of what is really happening. A macroscopic object may appear to have a velocity of zero relative to another object, but if you look closely enough you may find this is not exactly correct. This is not because a different definition of zero is being used, it is because of limitations in our ability to measure and detect things.

71

The “apple”, as used in the Newton story (possibly apocryphal?) is a unit of enlightenment. There is a long established history of units of enlightenment landing while one sits under a tree.

72

It’s not relevant to the point, though. In the math example, “apple” is taken as an abstract quantity. It is a stand-in for the number “one”. But the unit could be anything.

Physics, on the contrary is not abstract… it is physical, as the name implies. Therefore, in physics, “1 apple minus 1 apple” is a nonsense statement without even trying to compute it.

If instead your example had stated “1 gram of apple minus 1 gram of apple is zero grams of apple”, this is acceptable in both math and physics, because that comparison is, well, apples to apples.

73

What’s wrong with apples as a unit? I mean, they’re not SI, but one can treat apples as a unit just fine. I can take the number of Calories per apple and the number of apples per bushel and the number of bushels of apples I have, and do a unit conversion to find how many Calories I have.

74

The SI Brochure section 5.4.2 and NIST Special Publication 811 section 7.5 say you can’t do that.

FWIW, I think the SI Brochure and NIST SP8111 are wrong about this.

75

I think this is why zero is so hard to apply to the natural universe. If your physics equation divides, for example, by 1, even if 1 isn’t exactly defined, it still yields a useful result. Dividing by zero causes problems.

76

Dividing by zero is a problem in “mathematics” as well. I put it in quotes because an equation in physics is just maths, it’s just using maths to get an understanding of the universe.

77

Derived units are fine in physics, but they have to be derived from standard units. You can express weights in newtons or pounds or tons. They’re convertible because they’re standard.

Apples aren’t standard. They can only be used as units if you’re provided an assumption that they are spheres of a certain diameter, mass, and caloric content. Or whatever attributes are necessary to get the result that you’re intended to compute.

78

Of all the kooky things in this thread, using apples as a unit is one of the least kooky. For over a century the unit “kilogram” was based on chunk of metals that varied in mass over time. Sure, that’s more standard and more stable than an apple, but that’s a question of how precise you need things, not whether it makes philosophical sense. When someone says that a Boeing 747 weighs as much as 80 elephants, one doesn’t start quibbling about the non-standard nature of the unit “elephant”, since for the purposes at hand, that unit is doing a fine job.

Toward the original topic:

That thread is just about a particular bit of math jargon, wherein “identically zero” and “zero” both mean the same “zero” but the first one is talking about a case where something is in some sense always zero instead of only under certain conditions.

“Zero” isn’t any different in physics. You mention this article:

It is worth distinguishing “zero” the mathematical concept (which means the same thing in physics as in math) from any particular physical system that might relate to zero in some way. There may be interesting things to say about how a system is related to zero, but the existence of those interesting things doesn’t mean “zero” as a concept is any different in physics.

If one wants to talk about interesting physical differences from zero, that’s one thing. But there’s no reason to throw out the definition of “zero” when doing so. The meat is in the differences, not the zero itself.

79

Do you think the definition of zero could be refined with respect to modern physics? GR refined Newtonian physics. A refined definition of zero might be necessary as our technology allows for more precise observations (e.g. CERN, JWST…)

80

If the mass of, let’s say a neutrino is non-zero, then it is not zero. The problem is not with the definition of zero.

81

No. Zero is zero. There is no difference in definitions and no need to have a different definition. Something is either mathematically zero or it is not.

Aside from taking an apple away from a different apple, which is definitely not a real world problem, what specific real world problems do you see with zero? How many unicorns are in my living room? Zero. What speed am I moving with respect to my chair? Close to, but not zero. If you have an apple on a table and then remove it how many whole apples are left? Zero. How much apple is left? Not quite zero because if you look closely there will be some little bits of apple remaining.

In what way would redefining zero in modern physics to just be a very small number be useful? What would a number smaller then new-zero be?

82

If I have one apple in my refrigerator fruit drawer, and I eat said apple, I have zero apples remaining in my fruit drawer. It doesn’t matter if no two apples are exactly alike. It’s an irrelevant fact and has no bearing on the situation.

Example: A teacher has 25 students. Three of them are absent today. She has 22 students who are present. The fact that no two students are alike doesn’t change that fact.

83

We’ve explained 12 different times why this isn’t necessary or possible. Can you help us understand how those explanations were lacking, that you continue to restate the question?

The best tool for this purpose is very tiny numbers, and the supply of those is more than adequate to this purpose. We can’t, needn’t, and mustn’t redefine zero to include very tiny numbers.

Unless of course we’re trying to explain things that float, such as very small rocks.

84

Quoting these posts, because neither got a response from the OP and yet seem incredibly pertinent: