Hydrogen…an atom of only one proton and one electron (maybe one or rarely two neutrons) can arguably be placed on more than one position on the Periodic Table:
If a number, object, physical process, etc. is small enough…that is, it’s lack of internal structure makes it so basic/trivial/degenerate, it will lack the distinguishing characteristics of larger numbers/complex processes.
Do my fellow Dopers know if this “rule” is in fact formulated in academia?
I do know that small numbers are treated unfairly, for example, when people say “There’s strength in numbers” and you think 0.186 could do the trick and then they say “No that’s not what I meant”, or when you’re told that a number of unsuccessful attempts have already been made to do what you’re trying to do, and you think probably the number is 2, and again with the “No, that’s not what I meant” trick.
AFAIK with real numbers the answer is no … there’s no real number too small to add or multiply, every one behaves exactly like every other one … if fact, there’s no real number that’s too small to have a real number even smaller, which is something of the basis for calculus …
One rule that comes to mind is that a “system”, for instance a thermodynamic system, a random variable, or some such, will roughly speaking have an entropy that increases with the number of states available, or, even more basically, small systems simply have fewer possible states or configurations. There are not too many graphs with 3 vertices, etc, so there is more room for combinatorial coincidences at low numbers.
No need to invoke infinitesimal numbers to consider real numbers with respect to scaling. If you take real numbers with respect to, say, addition, and a is a scaling factor, you have a(x + y) = ax + ay.
One thing you can say is that if you have a locally compact topological group, then up to a multiplicative constant there is a unique invariant measure on the group. You can still scale it, though.
Its a boundary condition, an edge condition. What is element 1, is 1 odd ? is 1 prime ?
Its common that the edge or boundary or start is not indicative of the behaviour of the system.
Although its not clear that Hydrogen has to be at the edge for every possible periodic table… it could be in the middle ? Also, its not the only metal that is also participating in covalent bonds. There’s actually degrees inbetween ionic and covalent, and somewhere down the middle, the oxides of the metals are nearly covalent.
But your misunderstanding of hydrogen is not important.
Your first terms may also be axioms… since thats the starting point.
If so , its connected with Godel’s incompletement theorems, which imply that you cannot prove the axiom is true… you’ve already assumed it to be true, you can only show that the set of axioms are consistent.
Er, first of all, that is Haar’s theorem, proved by Alfréd Haar.
More importantly, that has to do with Chronos’s remark about the real numbers having no scale, and dimensional analysis as in physics, not to the original question about small numbers (to which I think the simple answer is that when you have more to work with you have more complexity, and vice versa).
Leo, I have no idea what you think you are on about, but personal attacks on other posters are not permitted in General Questions. No warning issued. However, I think it would be better that you not post further in this thread in order to avoid further misunderstandings.
I had the same thought - it seemed to me that the whole “law” idea was at least to some extent tongue-in-cheek, and that he joined in enthusiastically with that theme of poking fun at over-serious laws.
There are all sorts of examples where things are different for one or a few small numbers. For example, all primes are odd–with one minor exception. More esoteric: the only symmetric group with an outer automorphism is the group of permutations of 6 things.
If that’s the case, I’ll apologize (with the observation that if a remark might be taken as an insult, it is always wise to add a smilie to indicate the intent).
The “law” in question states: “There aren’t enough small numbers to meet the many demands made of them.” Which is going to result in your seeing patterns among small integers that aren’t really there, because there are so many different things happening there. (E.g. even a sequence that gets really big really fast - the factorials - has five terms in the sequence that are under 25.)
Yeah, there’s only one lousy prime that’s divisible by 2. Not only that, but there’s only one prime divisible by 3, only one prime divisible by 5, only one prime divisible by 7… (And no primes at all divisible by 4, or 6.)